TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 00, Number 0, Pages 000–000
S 0002-9947(XX)0000-0
INFINITE DIMENSIONAL PERFECT SET THEOREMS
TAMÁS MÁTRAI
Abstract. What largeness and structural assumptions on A ⊆ [R]ω can guarantee the existence
of a non-empty perfect set P ⊆ R such that [P ]ω ⊆ A? Such a set P is called A-homogeneous.
We show that even if A is open, in general it is independent of ZFC whether for a cardinal κ,
the existence of an A-homogeneous set H ∈ [R]κ implies the existence of a non-empty perfect
A-homogeneous set.
On the other hand, we prove an infinite dimensional analogue of Mycielski’s Theorem: if A is
large in the sense of a suitable Baire category-like notion then there exists a non-empty perfect Ahomogeneous set. We introduce fusion games to prove this and other infinite dimensional perfect
set theorems.
Finally we apply this theory to show that it is independent of ZFC whether Tukey reductions
of the maximal analytic cofinal type can be witnessed by definable Tukey maps.
1. Introduction
The Perfect Set Theorem says that an analytic subset of a Polish space is either countable or
has a non-empty perfect subset (see e.g. [11, Theorem 29.1 p. 226]). The complexity assumption
in this result is consistently optimal: in L there exists an uncountable Π11 set without non-empty
perfect subsets (see e.g. [9, Corollary 25.37]).
However, one is often obliged to quest for a perfect set which satisfies multidimensional relations.
Let N < ω be fixed and let [X]N denote the set of N element subsets of X. Then an N dimensional perfect set theorem should address the following problem: what largeness and structural
assumptions on A ⊆ [X]N can guarantee the existence of a non-empty perfect set P ⊆ X such that
[P ]N ⊆ A? Such a set P is called A-homogeneous.
In [12, Theorem 2.2 p. 620], W. Kubiś proved that if X is a Polish space, A ⊆ [X]N is Gδ and there
exists an uncountable A-homogeneous set then there exists of a non-empty perfect A-homogeneous
set. Obviously, as far as Gδ sets are concerned, this result is the exact multidimensional analogue
of the Perfect Set Theorem. The surprising fact is that the complexity assumption in this result is
also optimal: the Turing reducibility relation on 2ω (see Definition 2.4) defines an Fσ set A ⊆ [2ω ]2
such that there exists an A-homogeneous set H ⊆ 2ω with |H| = ω1 but there is no non-empty
perfect A-homogeneous set.
For A analytic, W. Kubiś and S. Shelah [13] investigated a rank function which decides whether
the existence of an A-homogeneous set with a given cardinality implies the existence of a non-empty
perfect A-homogeneous set (see also [8], [22]). They obtain in particular that for every α < ω1 , it is
2000 Mathematics Subject Classification. Primary 03E15.
The research was partially supported by the OTKA grants K 61600, K 49786 and K 72655 and by the NSERC
grants 129977 and A-7354.
c
XXXX
American Mathematical Society
1
2
TAMÁS MÁTRAI
consistent with ZFC that there exists an Fσ set A ⊆ [2ω ]2 such that there exists an A-homogeneous
set of cardinality ℵα but there is no non-empty perfect A-homogeneous set.
Nevertheless, the most frequently applied perfect set theorem is a classical result of J. Mycielski,
which says that if X is a non-empty perfect Polish space and AN ⊆ [X]N (N < ω) are co-meager
relations then there is a non-empty perfect set P ⊆ X such that [P ]N ⊆ AN (N < ω). So in
particular, if A ⊆ [X]N is co-meager then there exists a non-empty perfect A-homogeneous set.
Obviously, the largeness assumption in this result is not optimal.
In the present paper we study the existence of non-empty perfect homogeneous sets for infinite
dimensional relations A ⊆ [X]ω . Unlike in the finite dimensional case, it is not obvious how to
topologize [X]ω , therefore we usually assume that A ⊆ X ω is symmetric, i.e. it is invariant under
permutations of coordinates in X ω ; and a set H ⊆ X is called A-homogeneous if the injective
sequences of H, ISω (H) = {(xn )n<ω ∈ H ω : xn 6= xm (n < m < ω)} are in A.
It is obvious that if A ⊆ X ω is closed and H ∈ [X]ω1 is A-homogeneous then clX (H) is also
A-homogeneous hence there exists a non-empty perfect A-homogeneous set. In Section 2 we show
that this complexity assumption is also optimal (see Theorem 2.1).
Theorem 1.1. Let κ be a cardinal and suppose that there exists an Fσ set A ⊆ [2ω ]2 such that there
exists an A-homogeneous set of cardinality κ but there is no non-empty perfect A-homogeneous set.
Then there exists a symmetric open set U ⊆ (2ω )ω such that there exists a U -homogeneous set of
cardinality κ but there is no non-empty perfect U -homogeneous set.
Thus in the infinite dimensional case, by the above mentioned result of W. Kubiś and S. Shelah,
even for open relations, it is consistent that the existence of a homogeneous set of large cardinality
does not imply the existence of a non-empty perfect homogeneous set (see Corollary 2.3).
On the other hand, Mycielski’s Theorem has an infinite dimensional
analogue. For a Polish
S
space X, consider the σ-ideal M generated by the sets of the form n<ω (Mn × X ω\(n+1) ) where
Mn ⊆ X n+1 (n < ω) are meager. As we will see, an easy application of Mycielski’s Theorem yields
that if A ⊆ X ω satisfies X ω \ A ∈ M then there exists a non-empty perfect A-homogeneous set (see
Theorem 4.1). A more involved task is to find sufficient conditions for X ω \ A ∈ M. In Section 4 we
provide such sufficient conditions. In particular, we will show the following (see Corollary 4.7.2).
Theorem 1.2. Let X be a Polish space, let A ⊆ X ω be co-analytic and suppose that there exists a
non-meager A-homogeneous set. Then there exists a non-empty perfect A-homogeneous set.
As a corollary, we obtain that in the iterated perfect set model, for every co-analytic set A ⊆ X ω if
there exists an A-homogeneous set of cardinality continuum then there exists a non-empty perfect
A-homogeneous set (see Theorem 4.9). Moreover, we also obtain that in Cohen extensions the
existence of a homogeneous set of sufficiently large cardinality implies the existence of a non-empty
perfect homogeneous set (see Theorem 4.10). Thus by combining these result and Theorem 1.1, in
Section 4 we will prove the following.
Theorem 1.3. Let 1 < α < ω1 be an ordinal. Then it is independent of ZFC whether for an open
set A ⊆ (2ω )ω , the existence of an A-homogeneous set of cardinality ℵα implies the existence of a
non-empty perfect A-homogeneous set.
The key result toward Theorem 1.2 is proved by using a game which is obtained as a fusion of
Banach-Mazur games played on higher and higher dimensional powers of X (see Definition 3.2).
In Section 3 we introduce this game and characterize the winning strategies of the players (see
Theorem 3.4). It seems that our procedure of taking fusions is applicable to a wide class of games
INFINITE DIMENSIONAL PERFECT SET THEOREMS
3
of descriptive set theory. As an illustration, in Section 3.2 we briefly study the fusion of Perfect
Set Property games (see Definition 3.13 and Corollary 3.18). Independently of our work, M. Sabok
[21] introduced and studied similar games. We discuss the relation between fusion games and some
games of [21] and [29] in the introduction of Section 3.
Our study of infinite dimensional perfect set theorems was motivated by the problem whether
Tukey reducibilities of the maximal analytic cofinal type Imax can be witnessed by definable Tukey
maps. In Section 5 we recall the relevant definitions and we show that this problem is independent
of ZFC (see Theorem 5.7).
Finally we would like to thank Pandelis Dodos, David H. Fremlin, Michael Hrušák, Arnold W.
Miller, Benjamin D. Miller, Marcin Sabok, Lajos Soukup, Juris Steprāns, Boban Veličković, William
A. R. Weiss and Jindřich Zapletal for the helpful discussions. We are indebted to Stevo Todorčević
for his overall support in our research. We gratefully acknowledge the hospitality of the Department
of Mathematics of Rutgers University, NJ.
2. Open relations
The main result of this section is the following slightly generalized version of Theorem 1.1. Recall
that for every set H and α ≤ ω, ISα (H) = {(xn )n<α ∈ H α : xn 6= xm (n < m < α)}; and for
A ⊆ X α , H ⊆ X is called A-homogeneous if ISα (H) ⊆ A.
For every α ≤ ω, let Sα denote the permutation group on α. A set A ⊆ X α is symmetric if for
every π ∈ Sα , (ak )k<α ∈ A implies (aπ(k) )k<α ∈ A.
Theorem 2.1. Let A ⊆ (2ω )2 be a symmetric Fσ set such that there is no non-empty perfect
A-homogeneous set. Then there is a symmetric open set U ⊆ (2ω )ω such that
(1) for every (xk )k<ω ∈ (2ω )ω , if |{xk : k < ω}| = ω and there are i, j < ω such that xi 6= xj
and (xi , xj ) ∈ A, then (xk )k<ω ∈ U ;
(2) there is no non-empty perfect U -homogeneous set.
In particular, by 1, if H ⊆ 2ω is A-homogeneous then H is U -homogeneous, as well.
Before proving Theorem 2.1, we need some preparation. A well-known theorem of F. Galvin
states that if X is a non-empty perfect Polish space and B ⊆ [X]2 has the Baire property then
either there is a non-empty perfect B-homogeneous set or there is a non-empty perfect ([X]2 \ B)homogeneous set (see e.g. [11, Theorem 19.7 p. 130]). We will use the following corollary.
Corollary 2.2. Let A ⊆ (2ω )2 be a symmetric Fσ set such that there is no non-empty perfect Ahomogeneous set. Then every non-empty perfect set P ⊆ 2ω has a non-empty perfect subset Q ⊆ P
such that IS2 (Q) ∩ A = ∅.
Proof. Apply Galvin’s Theorem with X = P and B = A ∩ [P ]2 .
We introduce some terminology and notation. Fix a metric d1 on 2ω . For every 0 < n < ω, let
dn denote the coordinate supremum metric on (2ω )n generated by d1 . With an abuse of notation,
we extend dn to the hyperspace by setting, for every A, B ⊆ (2ω )n ,
dn (A, B) = inf{dn (a, b) : a ∈ A, b ∈ B}.
For every 0 < n < ω, (xk )k<n ∈ (2ω )n and δ > 0, let
B + ((xk )k<n , δ) = {(yk )k<ω ∈ (2ω )ω : ∃π ∈ Sω (dn ((xk )k<n , (yπ(k) )k<n ) < δ)}.
Then the sets B + ((xk )k<n , δ) are symmetric open subsets of (2ω )ω .
4
TAMÁS MÁTRAI
For every 1 < n < ω and (xk )k<n ∈ (2ω )n , we define δ((xk )k<n ) = min{|xi − xj | : i < j < n}.
For every 0 < n < ω, ∆n = {(x, . . . , x) ∈ (2ω )n : x ∈ 2ω }.
Proof of Theorem 2.1. Let Fn ⊆ (2ω )2 (n < ω) be symmetric closed sets such that
[
(2.1)
d2 (Fn , ∆2 ) > 0 (n < ω) and A \ ∆2 =
Fn .
n<ω
For every n < ω, set
[
Un = {B + ((xk )k<n+3 , δ((xk )k<n+3 )/4) : (xk )k<n+3 ∈ (2ω )n+3 , (x0 , x1 ) ∈ Fn }.
S
We show that U = n<ω Un fulfills the requirements.
By definition, U is a symmetric open set. To see 1, let (xk )k<ω ∈ (2ω )ω be such that |{xk : k <
ω}| = ω and there are i, j < ω such that xi 6= xj and (xi , xj ) ∈ A; say (xi , xj ) ∈ Fn . Since U
is symmetric, to have (xk )k<ω ∈ U it is enough to show that (xπ(k) )k<ω ∈ U for some π ∈ Sω .
That is, we can assume (x0 , x1 ) ∈ Fn and (xk )k<n+3 ∈ ISn+3 (2ω ). Then δ((xk )k<n+3 ) > 0 hence
B + ((xk )k<n+3 , δ((xk )k<n+3 )/4) 6= ∅. Thus (xk )k<ω ∈ Un ⊆ U , as required.
It remains to prove 2. Let P ⊆ 2ω be an arbitrary non-empty perfect set. By Corollary 2.2,
there is a non-empty perfect set Q ⊆ P such that IS2 (Q) ∩ A = ∅. We define inductively a sequence
(qk )k<ω ∈ ISω (Q) such that (qk )k<ω ∈
/ U ; this will complete the proof.
By induction on n < ω, we define qn ∈ Q (n < ω) and εn > 0 (0 < n < ω) such that
(i) for every i, j < ω, i 6= j implies qi 6= qj ,
(ii) for every 0 < n < ω, εn = min{d2 ((qi , qj ), Fk )) : i, j, k ≤ n},
(iii) for every 0 < i < n < ω, d1 (qn , qi ) < εi /2.
Let q0 , q1 ∈ Q be arbitrary satisfying q0 6= q1 . Let 0 < n < ω and suppose that qi (i ≤ n) and εi
(0 < i < n) are defined such that (i-iii) hold. In accordance with (iii), set
εn = min{d2 ((qi , qj ), Fk )) : i, j, k ≤ n}.
Then εn > 0 by IS2 (Q) ∩ A = ∅ and (2.1).
To satisfy (i) and (iii), let qn+1 ∈ Q \ {qi : i ≤ n} be arbitrary satisfying d1 (qn+1 , qi ) < εi /2
(i ≤ n); by the inductive assumption (iii) for n, such a qn+1 exists, namely any qn+1 ∈ Q\{qi : i ≤ n}
sufficiently close to qn fulfills the requirements.
Suppose (qk )k<ω ∈ U , say (qk )k<ω ∈ Un . Then (qk )k<ω ∈ B + ((xk )k<n+3 , δ((xk )k<n+3 )/4) for
some (xk )k<n+3 ∈ (2ω )n+3 such that (x0 , x1 ) ∈ Fn . Set δ = δ((xk )k<n+3 )/4. Then there are
k0 < k1 < · · · < kn+2 < ω and an enumeration n + 3 = {li : i < n + 3} such that
(iv) d1 (qki , xli ) < δ/4 (i < n + 3);
(v) for some a < b < n + 3, (xla , xlb ) ∈ Fn .
We distinguish two cases. Suppose first b < n + 2. By (iv),
(2.2)
d1 (qka , xla ) <
δ
δ
, d1 (qkb , xlb ) < .
4
4
We have 0 < kn+1 < kn+2 . So by (iii), d1 (qkn+2 , qkn+1 ) < εkn+1 /2. By b < n + 2 we have kb ≤ kn+1 ;
in addition we have n ≤ kn+1 , so εkn+1 ≤ d2 ((qka , qkb ), Fn ). By (2.2) and (v), d2 ((qka , qkb ), Fn ) <
δ/4. So to summarize,
d1 (qkn+2 , qkn+1 ) <
εkn+1
d2 ((qka , qkb ), Fn )
δ
≤
< .
2
2
8
INFINITE DIMENSIONAL PERFECT SET THEOREMS
5
Again by (iv),
d1 (qkn+1 , xln+1 ) <
δ
δ
, d1 (qkn+2 , xln+2 ) < ,
4
4
so the triangle inequality yields
δ
d1 (xln+1 , xln+2 ) ≤ d1 (xln+1 , qkn+1 ) + d1 (qkn+1 , qkn+2 ) + d1 (qkn+2 , xln+2 ) < 5 .
8
This contradicts the definition of δ.
Finally suppose b = n + 2; as in the previous case, we estimate d1 (xln+1 , xln+2 ). By (iv),
(2.3)
d1 (qka , xla ) <
δ
δ
δ
, d1 (qkn+1 , xln+1 ) < , d1 (qkn+2 , xln+2 ) < .
4
4
4
So by the triangle inequality,
δ
+d1 (qkn+1 , qkn+2 ).
2
< kn+2 . So by (iii), d1 (qkn+2 , qkn+1 ) < εkn+1 /2. Since ka ≤ kn+1 and
≤ d2 ((qka , qkn+1 ), Fn ). By (2.3),
(2.4) d1 (xln+1 , xln+2 ) ≤ d1 (xln+1 , qkn+1 )+d1 (qkn+1 , qkn+2 )+d1 (qkn+2 , xln+2 ) <
We have 0 < kn+1
n ≤ kn+1 , we get εkn+1
d2 ((qka , qkn+1 ), Fn ) ≤ d2 ((qka , qkn+1 ), (xla , xln+2 )) ≤
δ
δ
, d1 (qkn+1 , xln+2 ) ≤ + d1 (qkn+1 , xln+2 ).
max
4
4
By the triangle inequality and (2.3),
δ
d1 (qkn+1 , xln+2 ) ≤ d1 (qkn+1 , qkn+2 ) + d1 (qkn+2 , xln+2 ) < d1 (qkn+1 , qkn+2 ) + .
4
To summarize,
d2 ((qka , qkn+1 ), Fn )
εkn+1
δ d1 (qkn+1 , xln+2 )
δ d1 (qkn+1 , qkn+2 ) δ
≤
≤ +
< +
+ ;
2
2
8
2
8
2
8
i.e. d1 (qkn+2 , qkn+1 ) < δ/2. By (2.4) we obtained d1 (xln+1 , xln+2 ) < δ, which again contradicts the
definition of δ.
d1 (qkn+2 , qkn+1 ) <
Corollary 2.3. For every α < ω1 , it is consistent with ZFC that there exists a symmetric open set
U ⊆ (2ω )ω such that there exists an U -homogeneous set of cardinality ℵα but there is no non-empty
perfect U -homogeneous set.
Proof. By [13, Corollary 5.13 p. 159] or [22, Theorem 1.13 p. 15], it is consistent with ZFC that
there exists an Fσ set A ⊆ [2ω ]2 such that there exists an A-homogeneous set of cardinality ℵα but
there is no non-empty perfect A-homogeneous set. So the statement follows from Theorem 2.1. As we mentioned in the introduction, the Turing reducibility relation is a ZFC example for an Fσ
set T ⊆ [2ω ]2 such that there exists an uncountable T -homogeneous set but there is no non-empty
perfect T -homogeneous set. For the sake of completeness, we recall (a simplified version of) this
relation and prove its above-mentioned properties.
Definition 2.4. For every j < ω, set Tj< = {(x, y) ∈ 2ω × 2ω : x(i) = y(2j+1 · i + 2j ) (i < ω)} and
Tj> = {(x, y) ∈ 2ω × 2ω : (y, x) ∈ Tj< }. Then the Turing reducibility relation T ⊆ 2ω × 2ω is defined
S
by T = j<ω (Tj< ∪ Tj> ).
6
TAMÁS MÁTRAI
Proposition 2.5. The relation T is symmetric and Fσ such that there exists an uncountable T homogeneous set but there is no non-empty perfect T -homogeneous set.
Proof. It is obvious that Tj< , Tj> (j < ω) are closed sets, so T is symmetric and Fσ . Observe that
for given (xj )j<ω ⊆ 2ω , the point y ∈ 2ω defined by y(2j+1 · i + 2j ) = xj (i) (i, j < ω) satisfies
(xj , y) ∈ Tj< ⊆ T (j < ω). Hence a straightforward transfinite recursion yields an uncountable
T -homogeneous set.
Finally let P ⊆ 2ω be a non-empty perfect set. Observe that for every j < ω, Tj> is the graph
of a function, in particular Tj> ∩ (P × P ) (j < ω) are meager in P × P . By symmetry, this yields
T ∩ (P × P ) is meager in P × P . Hence P cannot be T -homogeneous, as required.
From T , by Theorem 2.1, we get a symmetric open set U ⊆ (2ω )ω with analogous properties.
In particular, this yields an example of an open set U ⊆ (2ω )ω which is dense even in the box
topology, still there is no non-empty perfect U -homogeneous set. We refer to Section 6.1 for a
further discussion of alternative topologies.
3. Fusion games
In this section we introduce the fusion of infinite sequences of games. The construction can
be performed for most of the usual games of descriptive set theory. However, the method of the
characterization of the winning strategies in a fusion game depends on the games whose fusion is
taken. Therefore, here we study only the fusion game of Banach-Mazur games in detail, which is
the most relevant for our perfect set theorems. In addition, in Section 3.2 we briefly discuss the
fusion game of Perfect Set Property games.
Independently of our work, M. Sabok [21] introduced and studied games which are very similar
to fusion games. The approach in [21], which originates from [29], makes explicit the connection between such games and iterated forcing. Fusion games, and the corresponding ideal M (see Definition
3.3) are also to be compared to the games and ideals of [29, Definition 5.1.1 p. 225].
3.1. Banach-Mazur games. Let X be a Choquet space such that there is a metric d on X whose
balls are open in X. The diameter of a set A ⊆ X is denoted by diamX (A). Recall that for every
A ⊆ X, in the Banach-Mazur game with payoff set A (see e.g. [11, Section 21.D p. 153]) two players
play
I:
U (0)
U (2)
...
U (2n)
...
II :
U (1)
U (3) . . .
U (2n + 1) . . .
where U (n) ⊆ X (n < ω) are non-empty open sets suchTthat U (n) ⊇ U (n + 1) and diam
T X (U (n)) <
2−n (n < ω), and player II wins the game if and only if i<ω U (i) is a singleton and i<ω U (i) ∈ A.
The following is well-known (see e.g. [11, Theorem 8.33 p. 51]).
Theorem 3.1. In the Banach-Mazur game with payoff set A,
(1) player I has a winning strategy if and only if there is a non-empty open set U ⊆ X such
that A ∩ U is meager;
(2) player II has a winning strategy if and only if X \ A is meager.
We define the fusion game of Banach-Mazur games. If Y is a set and s, t ∈ Y <ω , |s| denotes the
length of s and s_ t stands for the sequence s(0) . . . s(|s| − 1)t(0) . . . t(|t| − 1). We write s v t if
s = t||s| . If T ⊆ Y <ω is a tree and n < ω, set levn (T ) = {t ∈ T : |t| = n} and [T ] = {η ∈ Y ω : η|n ∈
INFINITE DIMENSIONAL PERFECT SET THEOREMS
7
T (n < ω)}. Recall also that the product space of Choquet spaces is Choquet (see e.g. [11, Exercise
8.13 p. 44]).
Definition 3.2. For every k < ω, let Xk be a Choquet
space such that there is a metric dk on
Q
Xk whose balls are open in Xk . For every A ⊆ k<ω Xk , Gω (A) denotes the fusion game of the
Banach-Mazur games with payoff set A, in which players I and II play
I : U0 (0)
II :
. . . (Ui (2(n − i)))i≤n
(U0 (2), U1 (0))
U0 (1)
...
(Ui (2(n − i) + 1))i≤n . . .
(U0 (3), U1 (1)) . . .
where for every k < ω, Uk (i) ⊆ Xk (i < ω) are non-empty open sets such that Uk (i) ⊇ Uk (i + 1)
−i
and
T diamXk (Uk (i)) < 2 (i <Tω), and player II wins the game if and only if for every k < ω,
i<ω Uk (i) is a singleton and ( i<ω Uk (i))k<ω ∈ A.
We denote by Gω the tree of partial runs of this game, ordered by end-extension. We set
nY
o
Uω =
Uk : n < ω, Uk ⊆ Xk is non-empty open (k < n) .
k<n
A quasi-strategy of player I is a non-empty pruned tree σ ⊆ Gω such that for every n < ω,
U ∈ lev2n+1 (σ) and u ∈ Uω , U _ u ∈ Gω implies U _ u ∈ σ.
Similarly, a quasi-strategy of player II is a non-empty pruned tree σ ⊆ Gω such that for every
n < ω, U ∈ lev2n (σ) and u ∈ Uω , U _ u ∈ Gω implies U _ u ∈ σ.
For every pruned tree σ ⊆ Gω , set
n
Y
Xk : ∃((Ui (2(k − i)))i≤k , (Ui (2(k − i) + 1))i≤k )k<w ∈ [σ]
W (σ) = (xk )k<ω ∈
k<ω
\
xk =
o
Uk (i) (k < ω) .
i<ω
For every P ∈ {I, II}, Σ(P
Q ) denotes the set of all quasi-strategies of player P in the game Gω ,
and we set W(P ) = {A ⊆ k≤n Xk : ∃σ ∈ Σ(P ) (W (σ) ⊆ A)}.
The winning strategies in Gω are characterized by the following Baire Category-like notion. For
arbitrary A ⊆ X × Y , we set PrX (A) = {x ∈ X : ∃y ∈ Y ((x, y) ∈ A)}.
Q
Definition 3.3. With the notation of Definition 3.2, for an arbitrary sequence Sn ⊆ k≤n Xk
(n < ω) we set
\ Y
Xk .
[(Sn )n<ω ] =
Sn ×
n<ω
n<k<ω
Q
We call U = (Un )n<ω an open tower if for every n < ω, Un ⊆ k≤n Xk is an open set and
Q
Un ∆ PrQk≤n Xk (Un+1 ) is meager in k≤n Xk . For two open towers U and V, U is dense in V if Un
Q
is dense in Vn (n < ω); and U is dense if Un is dense in k≤n Xk (n < ω). An open tower U is
non-empty if U0 6= ∅.
We define
n
Y
Y
M= M ⊆
Xk : ∃Mn ⊆
Xk (n < ω)
k<ω
k≤n
Mn is meager in
Y
k≤n
Xk (n < ω), M ⊆
[ n<ω
Mn ×
Y
n<k<ω
Xk
o
.
8
TAMÁS MÁTRAI
We discuss the relation of these notions to usual topologies before Lemma 3.7. However, note
that at this point it is not obvious that if U is a non-empty open tower then [U] 6= ∅. We will point
out this corollary of Xk (k < ω) being Choquet after the proof of Proposition 3.9.
We are in position to state the characterization of the winning strategies in Gω . Notice its analogy
with Theorem 3.1.
Q
Theorem 3.4. With the notation of Definition 3.3, for every A ⊆ k<ω Xk
(1) the following are equivalent:
(a) A ∈ W(I);
(b) there is a non-empty open tower U such that [U] ⊆ A;
(c) there is a non-empty open tower U such that [U] \ A ∈ M.
(2) the following are equivalent:
(a) A ∈ W(II);
(b) there
Q is a dense open tower U such that [U] ⊆ A;
(c) ( k<ω Xk ) \ A ∈ M.
We will need that Gω (A) is determined for co-analytic A, as well. To this end, we introduce an
unfolding of Gω , as follows.
Q
Definition 3.5. With the notation of Definition 3.2, for every F ⊆ ( k<ω Xk )×ω ω , Gω? (F ) denotes
the game with payoff set F in which players I and II play
I : U0 (0), y0
II :
. . . (Ui (2(n − i)))i≤n , yn
(U0 (2), U1 (0)), y1
U0 (1)
(U0 (3), U1 (1)) . . .
...
(Ui (2(n − i) + 1))i≤n . . .
where for every n < ω, ((Ui (2(k − i)))i≤k , (Ui (2(k
T − i) + 1))i≤k )k≤n ∈ Gω and yn ∈ ω, and player I
wins the game if and only if for every k < ω, i<ω Uk (i) is a singleton and
\
Uk (i)
, (yk )k<ω ∈ F.
k<ω
i<ω
For every pruned tree σ ⊆ Gω? , set
n
Y
W (σ) = ((xk )k<ω , (yk )k<ω ) ∈
Xk × ω ω :
k<ω
o
\
∃((Ui (2(k − i)))i≤k , yk , (Ui (2(k − i) + 1))i≤k )k<w ∈ [σ] xk =
Uk (i) (k < ω) .
i<ω
?
For every P ∈ {I, II}, Σ (P ) denotes the set of all quasi-strategies of player P in the game Gω? . We
set
n
Y
o
Xk × ω ω : ∃σ ∈ Σ? (I) (W (σ) ⊆ F )
W ? (I) = F ⊆
k<ω
n
Y
o
W ? (II) = F ⊆
Xk × ω ω : ∃σ ∈ Σ? (II) (W (σ) ∩ F = ∅) .
k<ω
For the winning strategies in the game
Gω?
we have the following.
Q
Theorem 3.6. With the notation of Definition 3.5, for every F ⊆ ( k<ω Xk ) × ω ω ,
(1) F ∈ W ? (I) implies PrQk<ω Xk (F ) ∈ W(I);
(2) F ∈ W ? (II) implies PrQk<ω Xk (F ) ∈ M.
INFINITE DIMENSIONAL PERFECT SET THEOREMS
9
It remains to prove our theorems.
Q We start with the analysis of open towers vs. M. It is obvious
that every M ∈ M is meager in k<ω Xk even in the boxQtopology. However, for an arbitrary
non-empty open tower U, [U] may also be nowhere dense in k<ω Xk in the box topology. Also, it
is easy to construct two non-empty open towers U and V such that [U] ∩ [V] is a singleton, i.e. such
an intersection may not contain a non-empty open tower. This indicates that open towers are not
the open sets of a carefully chosen topology. As the next lemma shows, “meager” and “nowhere
dense” coincide for M. In particular, once we obtained [U] 6= ∅ for every non-empty open tower U,
from Corollary 3.8 we get [U] ∈
/ M for every non-empty open tower U.
Lemma 3.7. With the notation of Definition 3.3, let U be a non-empty open tower and for every
n < ω, let Bn ⊆ Un be co-meager in Un . Then there is a non-empty open tower V = (Vn )n<ω such
that V is dense in U and [V] ⊆ [(Bn )n<ω ].
Q
Proof. For every n < ω, let Rn (i) ⊆ k≤n Xk (i < ω) be closed nowhere dense sets such that
S
Un \ Bn ⊆ i<ω Rn (i). Set
Y
[
Vn = Un \
Rn−i (i) ×
(n < ω).
Xk
i≤n
n−i<k≤n
It is obvious that Vn ⊆ Un is an open set which is dense in Un (n < ω). We have (xi )i<ω ∈ [V] if
and only if (xi )i≤n ∈ Vn (n < ω). So for every n < ω, by (xi )i≤n ∈ Vn we have (xi )i≤n ∈ Un , and
by (xi )i≤n+k ∈ Vn+k (k < ω) we have (xi )i≤n ∈
/ Rn (k) (k < ω). Thus (xi )i≤n ∈ Bn (n < ω), which
completes the proof.
Q
Corollary 3.8. With the notation of Definition 3.3, let A ⊆ k<ω Xk be arbitrary and let U be a
non-empty open tower. Then
(1) [U] \ A ∈ M if and only if there is a non-empty open tower V such that V is dense in U and
[V]
Q ⊆ A.
(2) ( k<ω Xk ) \ A ∈ M if and only if there is a dense open tower U such that [U] ⊆ A.
Q
S
Q
Proof. For 1, let first [U] \ A ⊆ n<ω (Mn × n<k<ω Xk ) where Mn ⊆ k≤n Xk (n < ω) are
meager. Then the statement follows from Lemma 3.7 applied with Bn = Un \ Mn (n < ω). The
converse is obvious.
Statement 2 is a special case of 1, so the proof is complete.
Q
Proposition 3.9. With the notation of Definition 3.3, let A ⊆ k<ω Xk be arbitrary.
(1) If there is a non-empty open tower U such that [U] ⊆ A then A ∈ W(I).
(2) If there is a dense open tower U such that [U] ⊆ A then A ∈ W(II).
Proof. Let U = (Un )n<ω . For 1, it is enough to construct a strategy σ ∈ Σ(I) such that W (σ) ⊆ [U].
We define σ ⊆ Gω by induction, as follows. Let (U0 (0)) ∈ σ if and only (U0 (0)) ∈ Gω and U0 (0) ⊆ U0 .
Let n < ω be arbitrary and suppose that σ ∩ lev2n+1 (Gω ) is already defined. For every U ∈
lev2n+1 (σ), Ui (2(n − i) + 1) (i ≤ n) and Ui (2(n + 1 − i)) (i ≤ n + 1), let
U _ ((Ui (2(n − i) + 1))i≤n ) ∈ σ and U _ ((Ui (2(n − i) + 1))i≤n , (Ui (2(n + 1 − i)))i≤n+1 ) ∈ σ
Q
if and only if U _ ((Ui (2(n − i) + 1))i≤n , (Ui (2(n + 1 − i)))i≤n+1 ) ∈ Gω and i≤n+1 Ui (2(n + 1 − i)) ⊆
Un+1 . This completes the inductive step of the definition of σ.
It is obvious that σ ⊆ Gω and that for every n < ω, U ∈ lev2n+1 (σ) and u ∈ Uω , U _ u ∈ Gω
implies U _ u ∈ σ. To see that σ is pruned, let n < ω, U ∈ lev2n+1 (σ) and (Ui (2(n − i) +
10
TAMÁS MÁTRAI
Q
1))i≤n be arbitrary such that U _ ((Ui (2(n − i) + 1))i≤n ) ∈ σ. Since i≤n Ui (2(n − i)) ⊆ Un and
Un ∆ PrQk≤n Xk (Un+1 ) is meager,
Y
Ui (2(n − i) + 1) × Xn+1 ∩ Un+1 6= ∅.
i≤n
In particular there are Ui (2(n + 1 − i)) (i ≤ n + 1) such that UiQ
(2(n + 1 − i)) ⊆ Ui (2(n − i) + 1)
(i ≤ n), diamXi (Ui (2(n + 1 − i))) < 2−(n+1−i) (i ≤ n + 1) and i≤n+1 Ui (2(n + 1 − i)) ⊆ Un+1 .
Thus U _ ((Ui (2(n − i) + 1))i≤n , (Ui (2(n + 1 − i)))i≤n+1 ) ∈ σ, which concludes σ ∈ Σ(I).
T
It remains to prove W (σ) ⊆ [U]. Let (xk )k<ω ∈ W (σ)Qbe arbitrary, say xk = i<ω Uk (i)
(k < ω) for some U ∈ [σ]. Then for every n < ω, (xk )k≤n ∈ i≤n Ui (2(n − i)) ⊆ Un . This shows
Q
(xk )k<ω ∈ Un × ( n<k<ω Xk ) (n < ω), so the statement follows.
Statement 2 follows by an analogous argument.
Since the spaces Xk (k < ω) are Choquet, in Gω both players can refine their quasi-strategies
in
T such a way that the resulting sequence ((Ui (2(k − i)))i≤k , (Ui (2(k − i) + 1))i≤k )k<ω satisfies
i<ω Uk (i) is a singleton for every k < ω. In particular, for every P ∈ {I, II}, A ∈ W(P ) implies
A 6= ∅. So by Proposition 3.9.1, if U is a non-empty open tower then [U] 6= ∅.
The proof of the following proposition closely follows the proofs of [11, Theorem 8.33 p. 51] and
[11, Theorem 21.8 p. 153].
Proposition 3.10. With the notation of Definition 3.3 and Definition 3.5,
(1) if A ∈ W(I) then there is a non-empty open tower U such that [U] ⊆ A;
(2) if A ∈ W(II) then there is a dense open tower U such that [U] ⊆ A;
(3) if F ∈ W ? (II) then PrQk<ω Xk (F ) ∈ M.
Proof. To see 1, let σ ∈ Σ(I) be such that W (σ) ⊆ A. We define a tree τ ⊆ σ by induction, as
follows. Let U0 ⊆ lev1 (σ) be a maximal family of pairwise disjoint open sets, and set lev1 (τ ) = U0 .
Let n < ω and suppose that lev2n+1 (τ ) is already defined. Let
Un+1 ⊆ {u ∈ Uω : ∃U ∈ lev2n+1 (τ ) ∃v ∈ Uω (U _ (v, u) ∈ σ)}
be a maximal family of pairwise disjoint open sets. For every u ∈ Un+1 fix one v(u) ∈ Uω such that
U _ (v(u), u) ∈ σ and set
lev2n+3 (τ ) = {U _ (v(u), u) ∈ σ : U ∈ lev2n+1 (τ ), u ∈ Un+1 }.
This completes the inductive step of the definition of τ . Observe that by requiring the members of
Un to be pairwise disjoint, for every n < ω and u ∈ Un there is a unique U ∈ lev2n+1 (τ ) such that
U (2n) = u. S
Let Un = Un (n < ω), we show that U = (Un )n<ω is a non-empty open tower and [U] ⊆ A. It
is obvious that U0 6= ∅. Let n < ω be arbitrary and let u ⊆ Un be an arbitrary non-empty open set;
we show (PrQk≤n Xk (Un+1 ))∩u 6= ∅. By passing to a subset we can assume u ⊆ u0 for some u0 ∈ Un .
Then there is a unique U ∈ lev2n+1 (τ ) such that U (2n) = u0 . By the definition of Gω , there is a
v ∈ Uω such that U _ v ∈ Gω and v ⊆ u. Since σ is a strategy of player I, there is a w ∈ Uω such
that U _ (v, w) ∈ σ, in particular PrQk≤n Xk (w) ⊆ v hence PrQk≤n Xk (w) ⊆ u. By the maximality
of Un+1 , there is a w0 ∈ Un+1 such that w ∩ w0 6= ∅. This shows (PrQk≤n Xk (Un+1 )) ∩ u 6= ∅. Since
Q
u ⊆ Un was arbitrary, we obtained Un \ PrQk≤n Xk (Un+1 ) is nowhere dense in k≤n Xk . Since
PrQk≤n Xk (Un+1 ) ⊆ Un follows immediately from the definition, we proved Un ∆ PrQk≤n Xk (Un+1 )
INFINITE DIMENSIONAL PERFECT SET THEOREMS
11
Q
is meager in k≤n Xk (n < ω). Thus U is a non-empty open tower. We also obtained that τ is a
pruned tree. Since τ ⊆ σ, we have W (τ ) ⊆ W (σ) ⊆ A, so it remains to see [U] ⊆ W (τ ).
Let (xk )k<ω ∈ [U] be arbitrary. For every n < ω, by Un being pairwise disjoint, there is a
unique v2n ∈ Un such that (xk )k≤n ∈ v2n . By the definition of τ , v2n ∈ Un means that there is a
Vn ∈ lev2n+1 (τ ) with Vn (2n) = v2n . But then (xk )k≤i ∈ Vn (2i), which implies Vn (2i) = v2i (i < n).
Thus there is a unique V ∈ [τ ] such that V (2n) = v2n (n < ω). This shows (xk )k<ω ∈ W (τ ), which
completes the proof of 1.
To see 2, let σ ∈ Σ(II) be such that W (σ)Q⊆ A. As in the proof of 1, we can define maximal
pairwise disjoint families Un of open subsets of k≤n Xk and a pruned tree τ ⊆ σ such that for every
S
n < ω and u ∈ Un there is a unique U ∈ lev2n+2 (τ ) satisfying U (2n+1) = u. Set UnQ= Un (n < ω).
Since σ is a strategy of player II, the maximality of Un implies Un is dense in k≤n Xk . Hence
U = (Un )n<ω is a dense open tower, and as in the proof of 1, we have [U] ⊆ W (τ ) ⊆ W (σ) ⊆ A.
To see 3, let σ ∈ Σ? (II) be such that W (σ) ∩ F = ∅. For every y ∈ ω <ω we say U ∈ Gω is
compatible with σ, y if |U | = 2|y| and (U (2i), y(i), U (2i + 1))i<|y| ∈ σ.
For every y ∈ ω <ω \ {∅} we construct a tree τy ⊆ Gω of height 2|y| such that
(i) every U ∈ lev2|y| (τy ) is compatible with σ, y;
(ii) {U (2|y| − 1) : U ∈ lev2|y| (τy )} is a family of pairwise disjoint open sets and for every U, V ∈
lev2|y| (τy ), U (2|y| − 2) 6= V (2|y| − 2) implies U (2|y| − 1) ∩ V (2|y| − 1) = ∅.
(iii) y v y 0 implies τy is the restriction of τy0 to sequences of length ≤ 2|y|;
(iv) τy is maximal with these properties.
Set τ∅ = ∅. Let y ∈ ω <ω \ {∅} be arbitrary, set y − = y||y|−1 and suppose that τy− is already
defined. Let
Uy ⊆ {u ∈ Uω : ∃U ∈ τy− ∃v ∈ Uω (U _ (v, u) is compatible with σ, y)}
be a maximal family of pairwise disjoint open sets. For every u ∈ Uy fix one v(u) ∈ Uω such that
U _ (v(u), u) is compatible with σ, y and set
lev2|y| (τy ) = {U _ (v(u), u) ∈ Gω : U ∈ lev2|y|−2 (τy− ), u ∈ Uy }.
This completes the inductive step of the definition of τy (y ∈ ω <ω \ {∅}).
It is obvious from the definition that (i-iv) hold.
S As in the proof
Q of statement 2, for every
y ∈ ω <ω \ {∅} the maximality of Uy implies Uy = Uy is dense in k<|y| Xk . Thus with Mn =
S Q
S
Q
{( k<|y| Xk ) \ Uy : y ∈ ω n+1 }, M = n<ω (Mn × n<k<ω Xk ) satisfies M ∈ M. It remains to
show that PrQk<ω Xk (F ) ⊆ M .
Q
Let ((xk )k<ω , (yk )k<ω ) ∈ ( k<ω Xk ) × ω ω be arbitrary and suppose (xk )k<ω ∈
/ M . Then in
particular, (xk )k≤n ∈ U(yk )k≤n (n < ω) so by (ii), there is a unique U ∈ [Gω ] such that U |2n+2 ∈
τ(yk )k≤n (n < ω). Then (U (2k), yk , U (k + 1))k<ω ∈ [σ], so ((xk )k<ω , (yk )k<ω ) ∈ W (σ) which implies
((xk )k<ω , (yk )k<ω ) ∈
/ F . This completes the proof.
Proof of Theorem 3.4. For 1, 1a ⇔ 1b follows from Proposition 3.9.1 and Proposition 3.10.1, while
1b ⇔ 1c is Corollary 3.8.1. For 2, 2a ⇔ 2b follows from Proposition 3.9.2 and Proposition 3.10.2,
and 2b ⇔ 2c is Corollary 3.8.2.
Proof of Theorem 3.6. For 1, if F ∈ W ? (I) then by omitting the yi s, player I gets a strategy in Gω
showing PrQk<ω Xk (F ) ∈ W(I). Statement 2 is Proposition 3.10.3.
The following is an immediate corollary of Theorem 3.4 and Theorem 3.6.
12
TAMÁS MÁTRAI
Q
Corollary
3.11. With the notation of Definition 3.2 and Definition 3.5, let A ⊆ k<ω Xk and
Q
F ⊆ ( k<ω Xk ) × ω ω be arbitrary.
Q
(1) If the game Gω (A) is determined then either ( k<ω Xk ) \ A ∈ M or there is a non-empty
open tower U such that A ∩ [U] = ∅.
(2) If the game Gω? (F ) is determined then either PrQk<ω Xk (F ) ∈ M or there is a non-empty
open tower U such that [U] ⊆ PrQk<ω Xk (F ).
The following is to be compared to [21, Proposition 3.29 p. 31] and [29, Section 5.1.3 p. 231].
In Proposition 4.8 we will show that the complexity assumptions in this result are consistently
optimal.
Corollary 3.12. With the notation of Definition 3.3,
Q
(1) if A ⊆ k<ω Xk is an analytic set then either A ∈ M or there is a non-empty open tower
U suchQ
that [U] ⊆ A.
Q
(2) if A ⊆ k<ω Xk is a co-analytic set then either ( k<ω Xk )\A ∈ M or there is a non-empty
open tower U such that A ∩ [U] = ∅.
Q
Proof. For 1, let F ⊆ ( k<ω Xk ) × ω ω be a closed set such that PrQk<ω Xk (F ) = A. Then the
game G ? (F ) is closed hence determined, i.e. the statement follows from Corollary 3.11.2. Statement
2 follows from 1 by taking complements.
We note that Corollary 3.12.1 has an alternative proof based on Borel determinacy, as follows.
Since the σ-ideal M is generated by closed sets, byQ[23, Theorem 1 p. 1023] either A ∈ M or there is
a Gδ set G ⊆ A such that G ∈
/ M. The game Gω (( k<ω Xk ) \ G) is determined, so we conclude that
there is a non-empty open tower U such that [U] ⊆ G ⊆ A. However, we believe that by avoiding
Borel determinacy and by presenting the unfolded game G ? we give a better insight to the fusion
game Gω .
3.2. Perfect Set Property games. For simplicity, in this section we assume X is a non-empty
perfect Polish space and we fix a countable base U in X which consists of non-empty open sets. We
study the following games.
Definition 3.13. Let 0 < N < ω be fixed. For every A ⊆ X N , PN (A) denotes the N -dimensional
Perfect Set Property game with payoff set A, in which two players play
I:
II :
((Uk (0, 0), Uk (0, 1)))k<N
...
(ik (0))k<N
...
((Uk (n, 0), Uk (n, 1)))k<N
...
(ik (n))k<N
...
where for every k < N , l < ω and j < 2 we have ik (l) ∈ {0, 1}, Uk (l, j) ∈ U, diamX (Uk (l, j)) <
2−l , Uk (l, 0) ∩T
Uk (l, 1) = ∅ and Uk (l + 1, j) ⊆ Uk (l,Tik (l)). Player I wins the game if and only if for
every k < N , l<ω Uk (l, ik (l)) is a singleton and ( l<ω Uk (l, ik (l)))k<N ∈ A.
INFINITE DIMENSIONAL PERFECT SET THEOREMS
13
Similarly, for every A ⊆ X ω , Pω (A) denotes the fusion game of the Perfect Set Property games
with payoff set A, in which two players play
I : (U0 (0, 0), U0 (0, 1))
((U0 (1, 0), U0 (1, 1)), (U1 (0, 0), U1 (0, 1)))
i0 (0)
II :
...
(i0 (1), i1 (0)) . . .
...
((Uk (n − k, 0), Uk (n − k, 1)))k≤n
...
(ik (n − k))k≤n
...
...
where for every k, l < ω and j < 2 we have ik (l) ∈ {0, 1}, Uk (l, j) ∈ U, diamX (Uk (l, j)) < 2−(k+l) ,
Uk (l, 0)T
∩ Uk (l, 1) = ∅ and Uk (l + 1, j) ⊆ Uk (l,
T ik (l)). Player I wins the game if and only if for every
k < ω, l<ω Uk (l, ik (l)) is a singleton and ( l<ω Uk (l, ik (l)))k<ω ∈ A.
Notice that P1 is the usual Perfect Set Property game (see e.g. [11, Section 21.A p. 149]). The
quasi-strategies of the players in the games PN (0 < N < ω) and Pω are defined analogously to
Definition 3.2. The characterization of existence of winning quasi-strategies involves the following
notions. Recall Ns = {σ ∈ 2ω : s v σ} (s ∈ 2<ω ).
Definition 3.14. Let 0 < N < ω be fixed. We call C = ((Uk (0), Uk (1)))k<N an
Q N -cube if for every
k < N and j < 2 we have Uk (l, j) ∈ U and Uk (0)∩Uk (1) = ∅. We define [C] = k<N (Uk (0)∪Uk (1)),
Q
and for every t ∈ 2N , C(t) = k<N Uk (t(k)). We set
diamX (C) = max{diamX (Uk (j)) : k < N, j < 2}.
N
We say that F ⊆ X is N -cube free if there is a δ > 0 such that for every N -cube C with
diamX (C) < δ there is a t ∈ 2N with F ∩ C(t) = ∅. We set
o
n
[
Fn ,
FN = F ⊆ X N : ∃Fn ⊆ X N (n < ω) Fn is N -cube free (n < ω), F ⊆
n<ω
n
Fω = F ⊆ X ω : ∃Fn ⊆ X n+1 (n < ω)
o
[ Fn is (n + 1)-cube free (n < ω), F ⊆
Fn × X ω\(n+1)
.
n<ω
ω N
N
A function f : (2 ) → X is cube preserving if for every n < ω and sk ∈ 2n (k < N ) there is
an N -cube C such that diamX (C) < 2−n and
h Y
i
f
N s_
= f (2ω )N ∩ C(t) (t ∈ 2N ).
t(k)
k
k<N
A function f : (2ω )ω → X ω is cube preserving if for every n < ω and sk ∈ 2n (k ≤ n) there is an
(n + 1)-cube C such that diamX (C) < 2−n and
h Y
i
ω\(n+1)
N s_
f
×
X
= f (2ω )ω ∩ C(t) × X ω\(n+1) (t ∈ 2n+1 ).
t(k)
k
k≤n
Notice that F1 = [X]≤ω . It is easy to see that every cube preserving function is continuous and
injective. The statement corresponding the Lemma 3.7 is the following.
Lemma 3.15. With the notation of Definition 3.14,
14
TAMÁS MÁTRAI
(1) F ⊆ X N is N -cube free if and only if clX N (F ) is N -cube free.
(2) FN (0n< N < ω) and Fω are σ-ideals generated
by closed sets. o
[ .
Fn × X ω\(n+1)
(3) Fω = F ⊆ X ω : ∃Fn ∈ Fn+1 (n < ω) F ⊆
n<ω
Proof. Since C(t) is open for every N -cube C and t ∈ 2N , the first statement follows. Then by
definition, 2 holds for FN (0 < N < ω) and Fω is generated by closed sets. To see 3 and that Fω is
a σ-ideal, observe that if F ⊆ X N is N -cube free then F × X M is N + M -cube free (M < ω). So
the statement follows by decomposing and re-indexing, as in the proof of Lemma 3.7.
Proposition 3.16. With the notation of Definition 3.13 and Definition 3.14,
(1) for every A ⊆ X N , player I has a winning quasi-strategy
in PN (A) if and only if there is
a cube preserving function f : (2ω )N → X N such that f (2ω )N ⊆ A.
(2) for every A ⊆ X ω , player I has a winning quasi-strategy
inPω (A) if and only if there is a
cube preserving function f : (2ω )ω → X ω such that f (2ω )ω ⊆ A.
Proof. For 1, suppose first player I has a winning quasi-strategy τ . By passing to a non-empty
pruned subtree we can assume that τ is a strategy, i.e. for every n < ω and s ∈ lev2n (τ ) there is a
unique N -cube C with s_ C ∈ τ . For every n < ω and sk ∈ 2n (k < N ) let C(sk )k<N be an N -cube
such that for every σk ∈ 2ω (k < N ),
(C(σk |m )k<N , (σk |m+1 )k<N )m≤n ∈ τ (n < ω);
(3.1)
this assignment is possible and unique since τ is a strategy. Moreover, diamX (C(sk )k<N ) < 2−n
((sk )k<N ∈ (2n )N , n < ω). Define f : (2ω )N → X N ,
\
C(σk |n )k<N .
f ((σk )k<N ) =
n<ω
n N
Then the cubes C(sk )k<N ((sk )k<N ∈ (2 ) , n < ω) witness that f is a cube preserving function.
Since τ is a winning strategy of player I, f (2ω )N ⊆ A follows.
Suppose now there is a cube preserving function f : (2ω )N → X N such that f (2ω )N ⊆ A, and
let C(sk )k<N ((sk )k<N ∈ (2n )N , n < ω) be the witnessing cubes. By
i
h Y
f (2ω )N ∩ [C(sk )k<N ] = f
Nsk = f (2ω )N ∩ C(sk |n−1 )k<N ((sk (n))k<N ),
k<N
we can assume
[C(sk )k<N ] ⊆ C(sk |n−1 )k<N ((sk (n))k<N ) ((sk )k<N ∈ (2n )N , 0 < n < ω).
Then the non-empty pruned tree τ defined by (3.1) is a winning quasi-strategy for player I.
Statement 2 follows by an analogous argument.
Proposition 3.17. With the notation of Definition 3.13 and Definition 3.14,
(1) for every A ⊆ X N , player II has a winning quasi-strategy in PN (A) if and only if A ∈ FN .
(2) for every A ⊆ X ω , player II has a winning quasi-strategy in Pω (A) if and only if A ∈ Fω .
Proof. For 1, suppose first player II has a winning quasi-strategy τ . Let U∅ = X N . For every
0 < n < ω and t ∈ lev2n (τ ), Ct = t(2n − 2) in an N -cube so we can define Ut = Ct (t(2n − 1)). Set
[
Ft = Ut \ {C(s) : C is an N -cube, s ∈ 2N , t_ C _ s ∈ τ } (t ∈ τ ).
INFINITE DIMENSIONAL PERFECT SET THEOREMS
15
We show that Ft (t ∈ τ ) are N -cube free. Fix t ∈ τ and let C = ((Uk (0), Uk (1)))k<N be
an arbitrary
N -cube with diamX (C) < 2−|t| ; we find s ∈ 2N such that Ft ∩ C(s) = ∅. Since
Q
Ut = k<N Vk for some Vk ∈ U (k < N ), we have
[C] ∩ Ut = [((Uk (0) ∩ Vk , Uk (1) ∩ Vk ))k<N ].
So since Ft ⊆ Ut , we can assume [C] ⊆ Ut . Then t_ C ∈ τ , hence there is an s ∈ 2N with t_ C _ s ∈ τ .
By definition, this implies Ft ∩ C(s) = ∅, as required.S
S
Since τ is countable, it remains to show A ⊆ t∈τ Ft . Suppose (xk )k<N ∈ X N \ t∈τ Ft .
N
By induction
T on n, we define N -cubes Cn and sn ∈ 2 (n < ω) such that ((Cn , sn ))n<ω ∈ [τ ] and
(xk )k<N ∈ n<ω Cn (sn ). Since τ is a winning quasi-strategy of player II, this implies (xk )k<N ∈
/ A,
as stated.
Since (xk )k<N ∈
/ F∅ , there are C0 , s0 with C0 _ s0 ∈ τ such that (xk )k<N ∈ C0 (s0 ). Let n < ω
be arbitrary and suppose Ci , si i ≤ n are defined such that ((Ci , si ))i≤n ∈ τ and (xk )k<N ∈ Cn (sn ).
Since (xk )k<N ∈ U((Ci ,si ))i≤n \ F((Ci ,si ))i≤n , there are Cn+1 , sn+1 with ((Ci , si ))i≤n_ Cn+1 _ sn+1 ∈ τ
and (xk )k<N ∈ Cn+1 (sn+1 ). This completes the inductive step of the definition of Cn , sn (n < ω),
and the proof of 1.
Statement 2 follows by an analogous argument.
Corollary 3.18. With the notation of Definition 3.13 and Definition 3.14,
(1) for every analytic set A ⊆ X N , either
A ∈ FN or there is a cube preserving function
f : (2ω )N → X N such that f (2ω )N ⊆ A.
(2) for every analytic set A ⊆ X ω , either A ∈ Fω or there is a cube preserving function
f : (2ω )ω → X ω such that f (2ω )ω ⊆ A.
Proof. To see 1, by Lemma 3.15.2, the σ-ideal FN is generated by closed sets. So by [23, Theorem
1 p. 1023], either A ∈ FN or there is a Gδ set G ⊆ A such that G ∈
/ FN . The game PM (G) is
determined, so the statement follows from Proposition 3.16.1 and Proposition 3.17.1.
Statement 2 follows by an analogous argument.
4. Infinite dimensional perfect set theorems
Our infinite dimensional perfect set theorems are based on the following easy observation. Recall
that for every set X and α ≤ ω, ISα (X) = {(xn )n<α ∈ X ω : xn 6= xm (n < m < α)}.
Theorem 4.1. Let X be a non-empty Choquet space such that X has no isolated points and there
is a metric d on X whose balls are open in X. Let A ⊆ X ω satisfy X ω \ A ∈ M. Then there is a
non-empty perfect set P ⊆ X such that ISω (P ) ⊆ A.
The proof of Theorem 4.1 uses the following version of Mycielski’s Theorem (see e.g. [20, Theorem
1 p. 141] and [11, Exercise 19.5 p. 130]).
Theorem 4.2. Let X be a non-empty Choquet space such that X has no isolated points and there
is a metric d on X whose balls are open in X. Let Mn ⊆ X n+1 (n < ω) be meager sets. Then there
is a non-empty perfect set P ⊆ X such that for every n < ω we have ISn+1 (P ) ∩ Mn = ∅.
S
Proof of Theorem 4.1. Let Mn ⊆ X n+1 (n < ω) be meager sets such that X ω \ A ⊆ n<ω (Mn ×
X ω\(n+1) ). By Theorem 4.2, there is a non-empty perfect set P ⊆ X such that for every n < ω we
have ISn+1 (P ) ∩ Mn = ∅. We show that P fulfills the requirements.
If (xi )i<ω ∈ X ω \ A then there is an n < ω such that (xi )i<ω ∈ Mn . Then (xi )i≤n ∈
/ ISn+1 (P )
hence (xi )i<ω ∈
/ ISω (P ), as required.
16
TAMÁS MÁTRAI
Corollary 3.12.2 say that for co-analytic A, X ω \ A ∈ M holds if there is no non-empty open
tower U such that [U] ∩ A = ∅. In the sequel we give various sufficient condition for this.
4.1. Largeness in category. Recall that for every Sn ⊆ X n+1 (n < ω) we have [(Sn )n<ω ] =
T
ω\(n+1)
). If n < ω, (xi )i≤n ∈ X n+1 and S ⊆ X n+2 then we set
n<ω (Sn × X
[S](xi )i≤n = {xn+1 ∈ X : (xi )i≤n+1 ∈ S}.
The most important additional property our topological spaces have to satisfy is the following.
Definition 4.3. A topological space X has the Kuratowski-Ulam property if for every n < ω and
for every meager set M ⊆ X n+2 ,
{(xi )i≤n ∈ X n+1 : [M ](xi )i≤n is non-meager in X}
is meager in X n+1 .
By [11, Theorem 8.41 p. 53], every second countable topological space has the Kuratowski-Ulam
property. In particular, all of our results hold for Polish spaces and for the canonical refinement of
Polish topologies turning a countable family of analytic sets into clopen sets (see e.g. [11, Theorem
25.18 p. 203]).
Our main technical notion is the following.
Definition 4.4. Let X be a topological space. We call W = (Wn )n<ω a flag if for every n < ω,
Wn ⊆ X n+1 and Wn = PrX n+1 (Wn+1 ). A flag W is of second category everywhere if W0 ⊆ X is
of second category everywhere and for every n < ω and (xi )i≤n ∈ Wn we have [Wn+1 ](xi )i≤n is of
second category everywhere in X.
If W is a flag and U is an open tower, then we say W is co-meager in U if U0 \ W0 is meager and
for every n < ω and (xi )i≤n ∈ Wn ∩ Un we have [Un+1 \ Wn+1 ](xi )i≤n is meager.
Lemma 4.5. Let X be a topological space and let U be a non-empty open tower. If X has the
Kuratowski-Ulam property then there is a flag W = (Wn )n<ω which is co-meager in U and Wn ⊆ Un
(n < ω). Moreover, if X is Polish then Wn (n < ω) can be taken Gδ .
Proof. We define Tn (i) ⊆ X n+1 (n, i < ω) by induction, as follows. Set Tn (0) = Un ∆ PrX n+1 (Un+1 )
(n < ω). Let i < ω and suppose that Tn (i) (n < ω) are defined. Then let
Tn (i + 1) = {(xi )i≤n ∈ X n+1 : [Tn+1 (i)](xi )i≤n is non-meager} (n < ω).
S
S
We show that Wn = Un \ i<ω k≤n (Tk (i) × X n−k ) (n < ω) fulfill the requirements.
It is obvious that Wn ⊆ Un (n < ω). Next we show that U0 \ W0 is meager and that for every
n < ω and (xi )i≤n ∈ Wn we have [Un+1 \ Wn+1 ](xi )i≤n is meager. Since U is an open tower, Tn (0)
(n < ω) are meager. Using the Kuratowski-Ulam property of X, it is easy to see that for every
n, i < ω, Tn (i) ⊆ X n+1 is meager. Hence U0 \ W0 is meager, as required.
Now let n < ω and let (xi )i≤n ∈ Wn . By (xi )i≤n ∈
/ Tn (j + 1) (j < ω) we have [Tn+1 (j)](xi )i≤n is
meager (j < ω). So [Un+1 \ Wn+1 ](xi )i≤n is meager, as required.
To see that W is a flag, we have to show Wn = PrX n+1 (Wn+1 ) (n < ω). Let first n < ω and
(xi )i≤n ∈ Wn be arbitrary. By (xi )i≤n ∈
/ Tn (0) we have [Un+1 ](xi )i≤n 6= ∅. As we have seen above,
[Un+1 \ Wn+1 ](xi )i≤n is meager, in particular [Wn+1 ](xi )i≤n 6= ∅, as required. While if n < ω and
S
S
(xi )i≤n+1 ∈ Wn+1 is arbitrary, then by (xi )i≤n+1 ∈ Un+1 and (xi )i≤n+1 ∈
/ i<ω k≤n+1 (Tk (i) ×
S
S
X n+1−k ) we get (xi )i≤n ∈ Un and (xi )i≤n ∈
/ i<ω k≤n (Tk (i) × X n−k ), i.e. (xi )i≤n ∈ Wn , as
required.
INFINITE DIMENSIONAL PERFECT SET THEOREMS
17
Finally if X is Polish, then Tn (0) (n < ω) are Fσ . By Montgomery’s Theorem (see e.g. [11,
Exercise 22.22 p. 174]), Tn (i) (n, i < ω) are Fσ , so by definition, Wn (n < ω) are Gδ .
Corollary 4.6. Let X be a topological space with the Kuratowski-Ulam property, and let U be a
non-empty open tower. Let V be flag which is of second category everywhere. Then [U] ∩ [V] 6= ∅.
Proof. By Lemma 4.5, there is a flag W which is co-meager in U and Wn ⊆ Un (n < ω). By
induction on n < ω, we define xn ∈ X (n < ω) such that (xi )i≤n ∈ Wn+1 ∩ Vn+1 (n < ω). Then by
(xn )n<ω ∈ [W] ∩ [V] and [W] ⊆ [U] the statement follows.
Since U0 6= ∅, U0 \ W0 is meager and V0 is of second category everywhere, we have W0 ∩ V0 6= ∅.
Let x0 ∈ W0 ∩ V0 be arbitrary.
Let n < ω and suppose that xi (i ≤ n) are defined such that (xi )i≤n ∈ Wn+1 ∩ Vn+1 . Since
W is a flag, we get [Wn+2 ](xi )i≤n 6= ∅. By Wn+2 ⊆ Un+2 this implies [Un+2 ](xi )i≤n 6= ∅. Moreover
[Un+2 \ Wn+2 ](xi )i≤n is meager , so since [Vn+2 ](xi )i≤n is of second category everywhere in X, we
have [Vn+2 ](xi )i≤n ∩ [Wn+2 ](xi )i≤n 6= ∅. Let xn+1 ∈ [Vn+2 ](xi )i≤n ∩ [Wn+2 ](xi )i≤n be arbitrary. This
completes the inductive step of the construction and finishes the proof.
Corollary 4.7. Let X be a non-empty Choquet space such that X has no isolated points, X has the
Kuratowski-Ulam property and there is a metric d on X whose balls are open in X. Let A ⊆ X ω .
(1) Suppose Gω (A) is determined and there is a H ⊆ X which is of second category everywhere
and ISω (H) ⊆ A. Then there exists a non-empty perfect set P ⊆ X such that ISω (P ) ⊆ A.
(2) Suppose A is co-analytic and there is a H ⊆ X which is non-meager and ISω (H) ⊆ A.
Then there exists a non-empty perfect set P ⊆ X such that ISω (P ) ⊆ A.
Proof. To see 1, set Vn = {(xi )i≤n ∈ H n+1 : xi 6= xj (i < j ≤ n)}; then V = (Vn )n<ω is a flag which
is of second category everywhere. Since [V] ⊆ A, by Corollary 4.6 we have A ∩ [U] 6= ∅ for every
non-empty open tower U. Hence by Corollary 3.11.1, X ω \ A ∈ M. So the statement follows from
Theorem 4.1.
By Corollary 3.12.2, statement 2 reduces to 1 by passing to a non-empty open subset of X where
H is of second category everywhere.
As we pointed out above, Corollary 4.7 is applicable if X is Polish or X is obtained from a
Polish space by turning a countable family of analytic sets into clopen sets the usual way. Even if
largeness in Baire category is not preserved during suchQ
refinement of topologies, this observation
shows that the game Gω (A) is informative for A = C ∩ k<ω Ak where C ⊆ X is co-analytic and
Ak ⊆ X (k < ω) are analytic.
Finally we show that the complexity assumptions in Corollary 3.12 are consistently optimal. The
assumption of the following proposition holds e.g. in L (see e.g. [9, Corollary 25.28 p. 495]).
Proposition 4.8. Assume there exists a Σ12 set D ⊆ 2ω which does not have the Baire property.
Then there exists a co-analytic set A ⊆ (2ω )ω such that A ∈
/ M but [U] 6⊆ A for every non-empty
open tower U.
Proof. By passing to a relative open subset we can assume D is non-meager and 2ω \ D is of second
category everywhere. Let C ⊆ 2ω × 2ω be a co-analytic set such that projection of C to the first
coordinate is D. Let
A = {(xi )i<ω ∈ (2ω )ω : (x0 , (xi (0))0<i<ω ) ∈ C}.
We show that A fulfills the requirements.
18
TAMÁS MÁTRAI
To see that A is co-analytic, let ϕ : ω × ω → ω × ω be a bijection defined by ϕ(0, i) = (0, i)
(i < ω), ϕ(i, 0) = (1, i − 1) (0 < i < ω) and ϕ : {(i, j) ∈ ω × ω : i · j 6= 0} → (ω \ {0, 1}) × ω
being any bijection. Then the automorphism of (2ω )ω induced by φ is a bijection between A and
C × (2ω )ω\{0,1} , hence A is co-analytic.
Observe that the projection of A to the first coordinate is D. If U is a non-empty open tower
then by Lemma 4.5, there exists a flag W = (Wn )n<ω which is co-meager in U and Wn ⊆ Un
(n < ω); in particular W0 6⊆ D. Since the projection of [W] to the first coordinate is W0 , [W] 6⊆ A
hence [U] 6⊆ A, as required.
To see A ∈
/ M, let M ∈ M be arbitrary. By Corollary 3.8.2, there is a dense open tower U
satisfying [U] ∩ M = ∅. So by Lemma 4.5, there exists a flag W = (Wn )n<ω which is co-meager in
U and Wn ⊆ Un (n < ω), in particular W0 is co-meager in 2ω and [W] ∩ M = ∅.
Let x0 ∈ W0 ∩ D be arbitrary and let x1 ∈ 2ω satisfy (x0 , x1 ) ∈ C. Set
Y
Un =
{x ∈ 2ω : x(0) = x1 (i)} (n < ω)
i≤n
ω n+1
and Vn = Wn+1 ∩ ({x0 } × (2 )
) (n < ω). Then U = (Un )n<ω is a non-empty open tower and
{x0 } × [U] ⊆ A; while V = (Vn )n<ω is a co-meager flag with {x0 } × [V] ⊆ [W]. By Corollary 4.6 we
have [U] ∩ [V] 6= ∅, i.e. A ∩ [W] 6= ∅ and so A 6⊆ M . This proves A ∈
/ M and completes the proof. 4.2. Largeness in cardinality. We show that in the iterated perfect set model and in Cohen
extensions the existence of a homogeneous set of sufficiently large cardinality implies the existence
of a non-empty perfect homogeneous set.
Theorem 4.9. Let V be a model obtained from a model of the Continuum Hypothesis by adding ω2
Sacks reals. Let A ⊆ Rω be a co-analytic set such that there is an A-homogeneous set of cardinality
ω2 . Then there exists a non-empty perfect A-homogeneous set.
Proof. Let H ∈ [R]ω2 be A-homogeneous. By [18, Theorem p. 581], there is a non-empty perfect
set X ⊆ R such that H ∩ X is of second category everywhere in X. Then the statement follows
from Corollary 4.7.2 applied to A ∩ X ω and H ∩ X.
For every cardinal µ, C[µ] = {f ⊆ µ × ω → 2 : |f | < ω} denotes the forcing for adding µ many
Cohen reals. We will use the elementary properties of the Cohen forcing stated in [9, Lemma 26.4
p. 514], [14, Lemma 2.2 p. 250] and [14, Theorem 2.1 p. 252] without further reference.
Theorem 4.10. In V , let κ = 2ℵ0 and let κ < λ ≤ µ be arbitrary cardinals. In V C[µ] , let A ⊆ (2ω )ω
be a co-analytic set such that there is an A-homogeneous set of cardinality λ. Then there exists a
non-empty perfect A-homogeneous set.
Proof. In V C[µ] , let R = {rα : α < µ} be the µ many Cohen reals added to V . First suppose
H ∈ [R]λ , i.e. that there is an I ∈ [µ]λ such that H = {rα : α ∈ I}.
S
In V C[µ] , let U = {U ⊆ 2ω : U open, |U ∩ H| < λ}. Set U = U. Then there is an IU ∈ [µ]≤ω
such that U ∈ V [{rα : α ∈ IU }]. Since {rα : α ∈
/ IU } are Cohen reals over V [{rα : α ∈ IU }], U
cannot be dense in 2ω . So we can find a non-empty open set O ⊆ 2ω \ U .
We show that in V C[µ] , every non-meager Gδ set G ⊆ O satisfies |G ∩ H| = λ. Given such a
G, let W ⊆ O be a non-empty open set such that G is dense in W . Let IG,W ∈ [µ]≤ω be such
that G, W ∈ V [{rα : α ∈ IG,W }]. Since W ⊆ O, by the definition of O we have |H ∩ W | = λ. The
set HG,W = {rα : α ∈ IG,W } is countable. So H ∩ W \ HG,W has cardinality λ, and each member
of this set is a Cohen real over V [{rα : α ∈ IG,W }]. Thus H ∩ W \ HG,W ⊆ G, so the statement
follows.
INFINITE DIMENSIONAL PERFECT SET THEOREMS
19
By Theorem 4.1, in order to conclude the existence of a non-empty perfect A-homogeneous set,
it is enough to show that Oω \ A ∈ M. By Corollary 3.12.2, this follows if we show that A ∩ [U] 6= ∅
for every non-empty open tower U = (Un )n<ω with Un ⊆ On+1 (n < ω). So let U be such an open
tower. By Lemma 4.5, there is a flag W = (Wn )n<ω which is co-meager in U and Wn (n < ω) are
Gδ . The proof will be complete if we show A ∩ [W] 6= ∅.
Let IW ∈ [µ]≤ω be such that W ∈ V [{rα : α ∈ IW }]. We define a sequence (hn )n<ω ∈ ISω (H)∩[W]
by induction, as follows. Since W0 ⊆ O is a non-meager Gδ set, we have |H ∩ W0 | = λ. Let
h0 ∈ H ∩ W0 be arbitrary.
Let n < ω be arbitrary and suppose that hi (i ≤ n) are defined such that (hi )i≤n ∈ ISn+1 (H) ∩
Wn . By definition, [Wn+1 ](hi )i≤n ⊆ O is a non-meager Gδ set. So |H ∩ [Wn+1 ](hi )i≤n | = λ, thus we
can pick hn+1 ∈ H ∩ [Wn+1 ](hi )i≤n \ {hi : i ≤ n}. This completes the inductive step of the definition
of (hn )n<ω and completes the proof of the special H ∈ [R]λ case.
In the general case, let H ∈ [2ω ]λ be A-homogeneous. For every h ∈ H fix an Ih ∈ [µ]≤ω such
that h ∈ V [{rα : α ∈ Ih }]. By the standard ∆-system argument and by extending V , we can assume
Ih ∩ Ih0 = ∅ (h, h0 ∈ H, h 6= h0 ). By passing to a subset of H, we can assume in addition that
tp(Ih ) = tp(Ih0 ) = η < ω1 (h, h0 ∈ H).
For every h ∈ H, let fh : (2ω )η → 2ω be a Borel function in V such that fh ((rα )α∈Ih ) = h. By
passing again to a subset of H, we can assume fh = fh0 = f (h, h0 ∈ H). Let X ⊆ (2ω )η be a
co-meager Gδ set such that f |X is continuous. Observe that for every h ∈ H, (rα )α∈Ih is a Cohen
real in (2ω )η , so in particular (rα )α∈Ih ∈ X (h ∈ H).
Set A = {(xk )k<ω ∈ Xω : (f (xk ))k<ω ∈ A ∩ ISω (2ω )}. As we observed above, H = {(rα )α∈Ih : h ∈
H} ⊆ X is an A-homogeneous set of Cohen reals of cardinality λ. Since A is co-analytic, by the
special case of Theorem 4.10 proved above, there exists a non-empty perfect A-homogeneous set
P ⊆ X.
Set P = f [P]. By definition, if (pk )k<ω ∈ ISω (P) then (f (pk ))k<ω ∈ ISω (2ω ). Hence f is
injective on P, i.e. by the continuity of f , P is a non-empty perfect set. Similarly, for every
(pk )k<ω ∈ ISω (P) we have (f (pk ))k<ω ∈ A, so P is A-homogeneous. This completes the proof. Proof of Theorem 1.3. Let 1 < α < ω1 be an ordinal. By [22, Claim 3.9 p. 39], it is consistent with
ZFC that there exists an Fσ set C ⊆ [2ω ]2 such that there exists a C-homogeneous set of cardinality
ℵα but there is no non-empty perfect C-homogeneous set. Then by Theorem 1.1, there exists a
symmetric open set U ⊆ (2ω )ω such that there exists a U -homogeneous set of cardinality ℵα but
there is no non-empty perfect U -homogeneous set.
On the other hand, by starting from a model with 2ℵ0 = ℵ1 and by adding ℵα+1 many Cohen
reals, we get a model in which 2ℵ0 = ℵα+1 and Theorem 4.10 implies the statement.
5. An application: definability of Tukey maps
In this section a set is called bounded if it is bounded from above. Similarly, directed means
upward directed. Let (P, ≤) and (Q, ≤) be directed partial orders. We say that (P, ≤) is Tukey
reducible to (Q, ≤), (P, ≤) ≤T (Q, ≤) in notation, if there is a function f : P → Q such that
for every unbounded set A ⊆ P , f [A] ⊆ Q is unbounded. Such an f is called a Tukey map.
If (P, ≤) ≤T (Q, ≤) and (Q, ≤) ≤T (P, ≤) then (P, ≤) and (Q, ≤) are called Tukey equivalent,
(P, ≤) ≡T (Q, ≤) in notation. In the sequel we do not write out the partial order when it is obvious
from the context.
An equivalent definition of Tukey reducibility is that P ≤T Q if and only if there is a function
g : Q → P such that for every cofinal set A ⊆ Q, g[A] ⊆ P is cofinal. This characterization indicates
20
TAMÁS MÁTRAI
that Tukey reductions provide information about the cofinal types of directed partial orders, and
explains why Tukey equivalence classes are also called cofinal types.
Tukey reducibility turns out to be the right tool for the comparison of cofinal types. Not only the
existence of a Tukey reduction between two directed partial orders relates many of their structural
properties, e.g. it is easy to see, using the equivalent definitions given above, that P ≤T Q implies
add(Q) ≤ add(P ) and cof(P ) ≤ cof(Q), but in addition, Tukey reductions account for many known
inequalities between cardinal invariants, e.g. all inequalities in the Cichoń diagram can be witnessed
by Tukey maps (see e.g. [5] and [1]). So a natural question arises: how many cofinal types of directed
partial orders are there?
The following result indicates that such a general endeavor has to face independence. As usual,
for every cardinal κ, [κ]<ω denotes the set of finite subsets of κ partially ordered by inclusion. We
remark that [κ]<ω is the maximal cofinal type of directed partial orders of cardinality ≤ κ (see e.g.
[28, Theorem 5.1 p. 13]).
Theorem 5.1. ([26, Theorem 9 p. 718])
(1) If the Continuum Hypothesis holds then there are 2ω1 many different cofinal types of directed
partial orders of cardinality ω1 .
(2) It is consistent with ZFC that {1, ω, ω1 , ω × ω1 , [ω1 ]<ω } are the only cofinal types of directed
orders of cardinality ≤ ω1 .
Therefore it is reasonable to restrict the cofinal diversity problem to classes of directed orders
which carry additional structures (see e.g. [5], [6]). One possible restriction is to assume definability
properties. Accordingly, in the present section we study analytic ideals on ω, i.e. such families
I ⊆ P(ω) which form an ideal under the partial order ⊆, and which are analytic subsets of P(ω),
endowed with the Cantor space topology. In Section 5.4 we will examine how restrictive this
assumption is (see Proposition 5.25 and Proposition 5.26).
Recall that in the definition of Tukey reducibility the reducing functions are not required to
possess any regularity properties. However, Tukey maps are not unique; e.g. if f is a Tukey map
and f ≤ f 0 pointwise then f 0 is also a Tukey map. So it is reasonable to ask whether a Tukey
reduction between analytic ideals can be witnessed by “nice” Tukey maps (see [5, Problem 3N (c)
p. 212], [16, Question 2 p. 193] and [27, Question 6.69] for analogous problems).
Problem 5.2. Let I, J ⊆ P(ω) be analytic ideals satisfying I ≤T J . Is there then a “definable”
Tukey map f : I → J ?
Depending on I and J , “definable” may mean continuous, Borel measurable, Souslin measurable (i.e. measurable with respect to the σ-algebra generated by analytic sets), Baire measurable,
Lebesgue measurable, etc. An affirmative answer to this problem could allow the use of descriptive
set theoretic methods for the study of an originally non-definable object.
Surprisingly, Problem 5.2 has an affirmative answer for many analytic ideals. In [24] the notion
of basic directed partial orders was introduced and the following result was proved.
Theorem 5.3. ([24, Theorem 5.3 p. 1890]) Let P, Q be basic directed partial orders satisfying
P ≤T Q. Then there is a Souslin measurable Tukey map f : P → Q.
We will recall the definition of basic directed partial orders in Section 5.4 (see Definition 5.27).
Here we only mention that every analytic P -ideal on ω is basic. However, there are many analytic
ideals on ω which are not basic in any topology; we will call such ideals non-basic. In [16, Section
7 p. 190] a sequence of non-basic Borel ideals was constructed which is strictly decreasing in the
Tukey hierarchy. In Section 5.4 we will prove the following.
INFINITE DIMENSIONAL PERFECT SET THEOREMS
21
Proposition 5.4. The structure (P(ω), ⊆? ) embeds into the family of non-basic Fσ ideals on ω
partially ordered by ≤T .
As we pointed out above, [2ℵ0 ]<ω is the maximal Tukey type among directed partial orders of
cardinality ≤ 2ℵ0 , so in particular among analytic ideals. As we will recall in Section 5.4, this
maximal cofinal type admits a representation as an Fσ ideal on ω.
Proposition 5.5. ([16, Proposition 3 p. 185]) There exists an Fσ ideal Imax ⊆ P(ω) such that
Imax ≡T [2ℵ0 ]<ω .
As we will see in Section 5.4, Imax is not basic; in particular, Theorem 5.3 does not apply to its
Tukey reductions. Therefore the following special case of Problem 5.2 is of particular interest (see
e.g. [27, Question 6.69]).
Problem 5.6. Let I ⊆ P(ω) be an analytic ideal satisfying Imax ≤T I. Is there then a “definable”
Tukey map f : Imax → I?
The purpose of this section is to show that even Problem 5.6 is independent of ZFC.
Theorem 5.7. Let I ⊆ P(ω) be an arbitrary analytic ideal.
(1) Let V be a model obtained from a model of the Continuum Hypothesis by adding ω2 Sacks
reals. Then in V , if Imax ≤T I then there is a continuous Tukey map f : Imax → I.
(2) If the Continuum Hypothesis holds then there is an analytic ideal J ⊆ P(ω) such that
Imax ≤T J, but if f : Imax → J is a Tukey map then f [Imax ] has no non-empty perfect
subsets. In particular, a Tukey map f : Imax → J cannot be Lebesgue measurable or have
the Baire property.
As a corollary, in Section 5.2 we obtain that it is consistent with ZFC that Imax has the primality
property (see Definition 5.13 and Corollary 5.15).
Presently, we do not know about any other special cases of Problem 5.2 where the same independence phenomenon appears. What makes Tukey reductions of Imax particularly easy to describe is
the following simple characterization.
Definition 5.8. Let (P, ≤) be a directed partial order. A set H ⊆ P is called strongly unbounded
if every A ∈ [H]ω is unbounded in (P, ≤).
Proposition 5.9. ([16, Section 1 p. 174]) Let (P, ≤) be a directed partial order and let κ be an
arbitrary cardinal. Then [κ]<ω ≤T P if and only if there exists a strongly unbounded set H ∈ [P ]κ .
5.1. A consistent positive answer to Problem 5.6. We will need the following simple observation.
Lemma 5.10. Let P and Q be directed partial orders and let f : P → Q be a Tukey map. If H ⊆ P
is strongly unbounded then f |H is finite-to-one and f [H] ⊆ Q is also strongly unbounded.
Proof. By definition, every A ∈ [H]ω is unbounded. Since f is a Tukey map, f [A] cannot be a
singleton, i.e. f is finite-to-one. The second statement immediately follows from the definition. The following implies Theorem 5.7.1, and also shows that the conclusion of Theorem 5.7.1 holds
for every projective ideal under suitable large cardinal assumptions.
Theorem 5.11. Let Γ be a projective pointclass such that every I ∈ Γ(P(ω)) has the Baire property
and for every A ∈ Γ(P(ω)ω ), the game Gω (A) is determined. Let κ ≤ 2ℵ0 be a cardinal such that
22
TAMÁS MÁTRAI
for every H ∈ [P(ω)]κ there exists a non-empty perfect set Q ⊆ P(ω) such that H ∩ Q is of second
category everywhere in Q. Let I ⊆ P(ω) be an ideal with P(ω) \ I ∈ Γ(P(ω)) such that [κ]<ω ≤T I.
Then there is a continuous Tukey reduction from [2ω ]<ω to I.
Proof. Let f : [κ]<ω → I be a Tukey map and let H = f [κ]. By Lemma 5.10, f is finite-to-one,
hence H has cardinality κ. So there exists a non-empty perfect set Q ⊆ P(ω) such that H ∩ Q is of
second category everywhere in Q. Since I ∩ Q has the Baire property in Q, by H ∩ Q ⊆ I ∩ Q there
is a co-meager
∩ Q ⊆ P(ω)satisfying H ∩ X is of second category everywhere in X.
Gδ set X ⊆ I S
Set A = (xi )i<ω ∈ X ω : i<ω xi ∈
/ I . Since the union function from P(ω)ω to P(ω) is Borel
and P(ω) \ I ∈ Γ(P(ω)), we have A ∈ Γ(X ω ). Then the game Gω (A) is determined and ISω (H ∩
X) ⊆ A, so by Corollary 4.7.1 there is a non-empty perfect set P ⊆ X such that ISω (P ) ⊆ A.
Then any continuous bijection f˜: 2ω → P is a continuous Tukey map from [2ω ]<ω to I.
Proof of Theorem 5.7.1. Recall that Imax ≡T [2ω ]<ω . By [18, Theorem p. 581], in the iterated
ℵ0
perfect set model for every H ∈ [2ω ]2 there exists a non-empty perfect set Q ⊆ P(ω) such that
H ∩ Q is of second category everywhere in Q. So by Corollary 3.12.2, the statement follows from
Theorem 5.11 with Γ = Π11 and κ = 2ℵ0 = ω2 .
5.2. Consistent primality of Imax . It is easy to see that the least upper bound in the Tukey
order of two ideals I, J ⊆ P(ω) is their direct sum, or disjoint union, defined as follows.
Definition 5.12. Let I, J ⊆ P(ω) be arbitrary ideals. Let E = {2n : n < ω}. Then I ⊕ J ⊆ P(ω)
is the ideal defined by
A ∈ I ⊕ J ⇔ {n < ω : 2n ∈ A ∩ E} ∈ I and {n < ω : 2n + 1 ∈ A \ E} ∈ J .
For a complete description of the cofinal types of analytic ideals, ideals having the following
primality property are of particular importance.
Definition 5.13. We say that an ideal I ⊆ P(ω) has the primality property if for every ideals
J , K ⊆ P(ω), I ≤T J ⊕ K implies I ≤T J or I ≤T K.
It is reasonable to ask which ideals have the primality property, and especially whether Imax
has the primality property (see e.g. [27, Question 6.68]). Note that by [26, Theorem 6 p. 715] the
primality of Imax fails among non-definable ideals. On the other hand, by [27, Theorem 6.71], the
primality of Imax holds for Souslin measurable Tukey reductions, as follows.
Theorem 5.14. ([27, Theorem 6.71]) Let I, J ⊆ P(ω) be analytic ideals such that there exists a
Souslin-measurable Tukey reduction of Imax to I ⊕ J . Then either Imax ≤T I or Imax ≤T J .
So by Theorem 5.7.1, we get the following.
Theorem 5.15. Let V be a model obtained from a model of the Continuum Hypothesis by adding
ω2 Sacks reals. In V , let I, J ⊆ P(ω) be analytic ideals such that Imax ≤T I ⊕ J . Then either
Imax ≤T I or Imax ≤T J .
Proof. By Theorem 5.7.1, there exists a continuous Tukey reduction of Imax to I ⊕ J . So the
statement follows from Theorem 5.14.
5.3. A consistent negative answer to Problem 5.6. In this section we construct the ideal J
of Theorem 5.7.2. To this end, the main technical step is to observe that it is enough to construct
an ideal of compact sets with the same properties.
INFINITE DIMENSIONAL PERFECT SET THEOREMS
23
Definition 5.16. Let K(2ω ) denote the space of compact subsets of the Cantor set endowed with
the Vietoris topology. A family I ⊆ K(2ω ) is called an ideal of compact sets if I is closed under
taking finite unions, and I is closed downward, i.e. for every K, L ∈ K(2ω ), K ⊆ L ∈ I implies
K ∈ I.
Throughout this section we use the following notation.
Definition 5.17. For every s ∈ 2<ω , set Ns = {x ∈ 2ω : s v x}. Let Ω = 2<ω . We define
Φ : K(2ω ) → P(Ω),
Φ(K) = {s ∈ Ω : Ns ∩ K 6= ∅}.
For every A ⊆ P(Ω), set A↓ = {B ∈ P(Ω) : ∃A ∈ A (B ⊆ A)}. For every I ⊆ K(2ω ) we define
I = (Φ[I])↓ .
Lemma 5.18. With the notation of Definition 5.17, we have the following.
(1) Φ is continuous, and for every K, L ∈ K(2ω ), K ⊆ L ⇔ Φ(K) ⊆ Φ(L). In particular,
Φ : K(2ω ) → Φ[K(2ω )] is a homeomorphism.
(2) If I ⊆ K(2ω ) is an ideal then I ⊆ P(Ω) is also an ideal.
(3) If A ⊆ P(Ω) is analytic then A↓ ⊆ P(Ω) is also analytic.
(4) If I ⊆ K(2ω ) is an analytic ideal then I ⊆ P(Ω) is also an analytic ideal.
(5) 3 and 4 also hold if we replace “analytic” with “Fσ ”.
Proof. It is obvious that for every K, L ∈ K(2ω ), K ⊆ L ⇔ Φ(K) ⊆ Φ(L). In particular, Φ is
injective. For every s ∈ Ω, Φ−1 ({A ∈ P(Ω) : s ∈ A}) = {K ∈ K(2ω ) : K ∩ Ns 6= ∅} is clopen in the
Vietoris topology, thus Φ is continuous. Since K(2ω ) is compact, Φ is a homeomorphism.
For 2, it is obvious that I is closed downward. To see that it is closed under taking unions,
pick Ai ∈ I (i < 2). Then there are Ki ∈ I (i < 2) such that Ai ⊆ Φ(Ki ) (i < 2). By 1,
A0 ∪ A1 ⊆ Φ(K0 ) ∪ Φ(K1 ) = Φ(K0 ∪ K1 ), so the statement follows.
To see 3, observe that {(A, B) ∈ P(Ω) × P(Ω) : B ⊆ A} is a closed relation, and A↓ is the
projection of {(A, B) ∈ P(Ω) × P(Ω) : B ⊆ A ∈ A} to the second coordinate. Hence A↓ is analytic
if A is analytic; and by P(Ω) being compact, A↓ is Fσ if A is Fσ .
Finally, 4 and 5 follow from 1, 2 and 3.
The ideals I and I are cofinally similar, as well.
Lemma 5.19. Let I ⊆ K(2ω ) be an analytic ideal. Then I ≡T I; moreover,
(1) Φ : I → I and Φ−1 : Φ[I] → I are continuous Tukey maps;
(2) there is a Tukey reduction Φ−1 ◦ f : I → I where f : I → I is a Souslin measurable Tukey
map such that H ⊆ f (H) ∈ Φ[I] (H ∈ I).
Proof. To see 1, let H ⊆ I be an arbitrary set. If Φ[H] ⊆ I is bounded then there is an L ∈ I
such that Φ(K) ⊆ Φ(L) (K ∈ H). By Lemma 5.18.1, this implies K ⊆ L (K ∈ H), i.e. H is also
bounded. Hence Φ is a Tukey map, which is continuous by Lemma 5.18.1. The argument for Φ−1
is identical.
For 2, consider the set P = {(A, B) ∈ I × Φ[I] : A ⊆ B}. Then P ⊆ P(Ω) × P(Ω) is analytic, so
by the Jankov-von Neumann Uniformization Theorem (see e.g. [11, Theorem 18.1 p. 120]) P has a
Souslin measurable uniformizing function f : I → Φ[I]. Also, H ⊆ f (H) ∈ Φ[I] (H ∈ I) holds.
Since f pointwise dominates the identity function, it is a Tukey map. Then by 1, Φ−1 ◦ f : I → I
is also a Tukey map, which completes the proof.
Corollary 5.20. Let I ⊆ K(2ω ) be an analytic ideal.
24
TAMÁS MÁTRAI
(1) For every cardinal κ we have [κ]<ω ≤T I ⇔ [κ]<ω ≤T I.
(2) There exists a non-empty perfect strongly unbounded subset of I if and only if there exists
a non-empty perfect strongly unbounded subset of I.
Proof. Statement 1 follows from Lemma 5.19. To see 2, suppose first that P ⊆ I is a non-empty
perfect strongly unbounded set. By Lemma 5.18.1, Φ is continuous and injective. So Φ[P ] is also
non-empty and perfect. By Lemma 5.19.1, Φ is a Tukey map, i.e. by Lemma 5.10, Φ[P ] is strongly
unbounded.
Finally let P ⊆ I be a non-empty perfect strongly unbounded set. Let f be the function of
Lemma 5.19.2. Then f |P is Souslin measurable, hence it has the Baire property, so by passing to
a non-empty perfect subset of P we can assume that f |P is continuous (see e.g. [11, Theorem 8.38
p. 52]).
By Lemma 5.10 and Lemma 5.19.2, f |P is finite-to one. Thus f [P ] ⊆ Φ[I] is an uncountable
compact strongly unbounded set, in particular it has a non-empty perfect subset Q, which is also
strongly unbounded. So by Lemma 5.19.1 and Lemma 5.10, Φ−1 (Q) ⊆ I is a non-empty perfect
strongly unbounded set.
Our main theorem is the following.
Theorem 5.21. There is an analytic ideal J ⊆ K(2ω ) with the following properties.
(1) [ω1 ]<ω ≤T J .
(2) [ω2 ]<ω 6≤T J .
(3) There is no non-empty perfect strongly unbounded subset of J .
The following corollary answers [16, Conjecture 2 p. 194] in the negative.
Corollary 5.22. There is an analytic ideal J ⊆ P(ω) with the following properties.
(1) [ω1 ]<ω ≤T J.
(2) [ω2 ]<ω 6≤T J.
(3) There is no non-empty perfect strongly unbounded subset of J.
Proof. Let J be the ideal of Theorem 5.21 and set J = (Φ[J ])↓ . By Lemma 5.18.4, J is an analytic
ideal. By Corollary 5.20, 1, 2 and 3 follows from Theorem 5.21.
Also, Theorem 5.7.2 easily follows.
Proof of Theorem 5.7.2. Let J be the ideal of Corollary 5.22. Assume the Continuum Hypothesis
holds. Then Imax ≡T [ω1 ]<ω ≤T J by Corollary 5.22.1.
Let f : Imax → J be an arbitrary Tukey map. By Lemma 5.10, f [Imax ] ⊆ J is strongly unbounded, so every subset of f [Imax ] is strongly unbounded, as well. Hence by Corollary 5.22.3,
f [Imax ] has no non-empty perfect subsets.
If f is either Lebesgue measurable or has the Baire property then there is an uncountable Borel
set B ⊆ Imax such that f |B is continuous. By Lemma 5.10, f [B] ⊆ J is an uncountable strongly
unbounded analytic set, in particular it has a non-empty perfect strongly unbounded subset. This
contradicts Corollary 5.22.3.
It remains to prove Theorem 5.21. Recall the definition of the Cantor-Bendixson derivative (see
e.g. [11, Definition 6.10 p. 33]): for every K ∈ K(2ω ), set K 0 = {x ∈ K : x ∈ cl2ω (K \ {x})}. We
define, by induction on n < ω, the iterated derivatives: K (0) = K, K (n+1) = (K (n) )0 (n < ω). Also
recall that there exists an analytic equivalence relation E ⊆ 2ω × 2ω such that E has exactly ω1
many equivalence classes (see e.g. [9, Exercise 32.3 p. 625]).
INFINITE DIMENSIONAL PERFECT SET THEOREMS
25
Definition 5.23. Set R = {K ∈ K(2ω ) : ∃n < ω (K (n) = ∅)}. Let E ⊆ 2ω × 2ω be an analytic
equivalence relation with exactly ω1 many equivalence classes. We define
(5.1)
L = {K ∈ K(2ω ) : K ∈ R, ∀x, y ∈ K ((x, y) ∈ E)}.
It is known that R is a Borel subset of K(2ω ) (see [3] and [4] for its exact Borel class). However,
due to the universal quantification in the definition of L, the following is not straightforward.
Proposition 5.24. L ⊆ K(2ω ) is analytic.
Proof. For every n < ω, S
let ϑn : K(2ω )n → K(2ω ) denote the union function, i.e. for every (Ki )i<n ∈
ω n
K(2 ) , ϑn ((Ki )i<n ) = i<n Ki . It is clear that ϑn (n < ω) are continuous.
ω
Recall Ns = {σ ∈ 2ω : s v σ} (s ∈ 2<ω ). We define Ln , L+
n ⊆ K(2 ) by induction on n, as
ω ≤1
follows. Set L0 = [2 ] . If n < ω and Ln is already defined, set
[
(5.2)
L+
ϑm [{(Li )i<m : ∀i < m ∃ui ∈ Li ((ui , uj ) ∈ E (i, j < m))}]
n = {∅} ∪
m<ω
and
(5.3)
Ln+1 = L0 ∪ {L ∈ K(2ω ) : ∃u, v ∈ L (u 6= v, (u, v) ∈ E, L \ Nu|m ∈ L+
n (m < ω))}.
ω
It follows from the definition that Ln , L+
ω) are analytic sets and Ln ⊆ L+
n ⊆ K(2
n (n < ω).
S ) (n <
+
So the proof will be complete if we show L = n<ω Ln .
S
+
(n+1)
= ∅ and
To see n<ω L+
n ⊆ L, we show by induction on n that for every L ∈ Ln we have L
(x, y) ∈ E (x, y ∈ L). The statement is obvious for n = 0; so let n < ω be arbitrary and suppose
+
that the statement holds for L+
n . Let L ∈ Ln+1 be arbitrary. Suppose first L ∈ Ln+1 , and let
u, v ∈ L be as in (5.3). By the inductive hypothesis, L(n) ⊆ {u} and (x, y) ∈ E (x, y ∈ L \ {u}).
Thus L(n+1) = ∅, and (u, v) ∈ E, u 6= v imply (x, y) ∈ E (x, y ∈ L).
S
(n+1)
=∅
In the general case, for some m < ω, let L = i<m Li where Li ∈ Ln+1 (i < m). Since Li
(n+1)
(i < m), we have L
= ∅. Let ui ∈ Li (i < m) be as in (5.2). Since (x, y) ∈ E (x, y ∈ Li , i < m)
and (ui , uj ) ∈ E (i, j < m), we get (x, y) ∈ E (x, y ∈ L), as required.
S
(n+1)
To see L ⊆ n<ω L+
= ∅ implies
n , we show by induction on n that for every L ∈ L, L
+
+
0
L ∈ Ln . For n = 0, L = ∅ holds if and only if L is finite; and then L ∈ L0 . So let n < ω and
suppose that the statement holds for n. Let L ∈ L be arbitrary satisfying L(n+2) = ∅. If L ∈ L0
then L ∈ Ln+1 follows. If L ∈
/ L0 and L(n+1) = ∅ then let u, v ∈ L be arbitrary with u 6= v.
By L ∈ L, we have (u, v) ∈ E. By L(n+1) = ∅ and by the inductive assumption, L \ Nu|m ∈ L+
n
(m < ω), so L ∈ Ln+1 .
If L(n+1) is a singleton then set {u} = L(n+1) and let v ∈ L \ {u} be arbitrary. As in the previous
case, by the inductive assumption, L \ Nu|m ∈ L+
n (m < ω), so L ∈ Ln+1 .
(n+1)
Finally if |L
| > 1 then it is a finite set, say L(n+1) = {ui : i < m} for some m < ω. Let
ω
Li ∈ K(2 ) (i < m) be arbitrary satisfying ui ∈ Li ⊆ L \ {uj : j < m, j 6= i} (i < m) and
S
(n+1)
L = i<m Li . Since Li
= {ui } (i < m), we have Li ∈ Ln+1 (i < m). By L ∈ L, (ui , uj ) ∈ E
+
(i, j < m) hence L ∈ Ln+1 , as stated.
Proof of Theorem 5.21. Recall that for every n < ω, ϑn : K(2ω )n → K(2ω ) denotes the union
function. With the set L defined in (5.1), set
[
J =
ϑn [Ln ].
n<ω
We show that J fulfills the requirements.
26
TAMÁS MÁTRAI
Since L is closed downward and J is closed under taking finite unions, J is an ideal of compact
sets. The functions ϑn (n < ω) are continuous hence by Proposition 5.24, J is analytic. It is
immediate from the definition that for every K ∈ K(2ω ), K ∈ J if and only if
(i) K intersects only finitely many equivalence classes of E;
(ii) for every equivalence class C ⊆ 2ω of E, K ∩ C ∈ K(2ω ) and (K ∩ C)(n) = ∅ for some n < ω.
Let H ∈ [2ω ]ω1 be an arbitrary set such that (x, y) ∈
/ E (x, y ∈ H) and set H = [H]1 . By (i),
H ⊆ J is strongly unbounded, i.e. by Proposition 5.9, 1 holds.
To see 2, by Proposition 5.9 we have to show that if H ∈ [J ]ω2 then H is not strongly unbounded.
So let H ∈ [J ]ω2 be arbitrary. Let {Eα : α < ω1 } be an enumeration of the equivalence classes of
E. By (i) and (ii), there are n < ω, A ∈ [ω1 ]<ω and H0 ∈ [H]ω2 such that for every K ∈ H0 ,
K ∩ Eα 6= ∅ ⇔ α ∈ A, and (K ∩ Eα )(n) = ∅ (α ∈ A).
Q
Consider the space X = α∈A (J ∩ K(Eα )) and the set
)
(
[
0
X = (Kα )α∈A : Kα ∈ J ∩ K(Eα ),
Kα ∈ H .
α∈A
Then X is a separable metric space and X ⊆ X is an uncountable set. Since X is Lindelöf, there is a
sequence (Kα (m))α∈AS∈ X (m < ω) such thatS
limm<ω Kα (m) = Kα
(Kα )α∈A ∈ X and an injective
S
(α ∈ A). Set H(m) = α∈A Kα (m) (m < ω), H = α∈A Kα and L = H ∪ n<ω H(m). Observe
that H ∈ J , in particular there is a d < ω such that H (d) = ∅. Also, H(m) ∈ H0 implies
H(m)(n) = ∅ (m < ω).
We have L ∩ Eα 6= ∅ ⇔ α ∈ A (α < ω1 ) and L ∩ Eα ∈ K(2ω ) (α ∈ A). Moreover, L(n) ⊆ H
hence L(n+d) = ∅. By (i) and (ii) this implies L ∈ J . Since ((Kα (m))α∈A )m<ω is injective,
{H(m) : m < ω} ⊆ H is an infinite set. By H(m) ⊆ L (m < ω), H is not strongly unbounded, as
stated.
Finally observe that by Shoenfield’s Absoluteness (see e.g. [9, Theorem 25.20 p. 490]), a nonempty perfect strongly unbounded subset of J would remain strongly unbounded in any forcing
extension. Hence 3 follows from 2.
5.4. Miscellanea. In the previous sections we restricted our investigations to analytic ideals I ⊆
(P(ω), ⊆). Unfortunately, there are simple directed partial orders which are not cofinally similar to
any such ideal. The following may be considered as a negative answer to [16, Question 1 p. 193].
Proposition 5.25. Let 1 denote the one element set with the trivial order, let ω denote the first
infinite ordinal with its usual well-order, and let (ω ω , ≤? ) be the set of all functions from ω to ω,
partially ordered by eventual dominance.
(1) If I ⊆ (P(ω), ⊆) is an arbitrary directed set then I ≤T 1 or ω ≤T I.
(2) The directed partial order (ω ω , ≤? ) satisfies ≤? ⊆ ω ω × ω ω is Fσ but it is incomparable with
ω in the Tukey order. In particular, (ω ω , ≤? ) 6≡T I for every directed set I ⊆ (P(ω), ⊆).
S
Proof. To see 1, let S = S I. If S ∈ I then I ≤T 1. If S ∈
/ I, let (Sn )n<ω ⊆ I be an increasing
sequence satisfying S = n<ω Sn ; such a sequence exists since I is directed. Then f : ω → I,
f (n) = Sn (n < ω) is a Tukey map.
For 2, ω 6≤T (ω ω , ≤? ) follows from the fact that every countable subset of (ω ω , ≤? ) is bounded.
To see (ω ω , ≤? ) 6≤T ω let f : (ω ω , ≤? ) → ω be an arbitrary function. Then there is an n < ω such
that f −1 ({n}) ⊆ ω ω is non-meager. Since for every ϕ ∈ ω ω , {ψ ∈ ω ω : ψ ≤? ϕ} is meager, such a
set cannot be bounded. Hence f is not a Tukey map. The second statement follows from 1.
INFINITE DIMENSIONAL PERFECT SET THEOREMS
27
On the other hand, closed directed partial orders on analytic spaces admit representations as
analytic ideals on ω.
Proposition 5.26. Let (I, ≤) be a topological space endowed with a directed partial order such that
I is a continuous image of a Polish space and ≤⊆ I × I is closed. Then there is an analytic ideal
I ⊆ (P(ω), ⊆) such that (I, ≤) ≡T I.
Proof. Let G ⊆ 2ω be an arbitrary set homeomorphic to ω ω , and let f : G → I be a continuous
surjection (see e.g. [11, Theorem 7.9 p. 38]). For every a ∈ I, set La = {b ∈ I : b ≤ a}. Note that
La ⊆ I (a ∈ I) are closed. We show that
I = {K ∈ K(2ω ) : ∃a ∈ I (K ⊆ cl2ω (f −1 (La ))}
is an analytic ideal of compact subsets of 2ω such that (I, ≤) ≡T I. Then, with the notation of
Definition 5.17, by Lemma 5.19, I = (Φ[I])↓ fulfills the requirements.
It is obvious that I is closed downward. Since I is directed, I is closed under taking finite unions.
Next we show that I is analytic. Let F(G) denote the set of closed subsets of G endowed with the
Effros Borel structure (see e.g. [11, Section 12.C p. 75]); recall that a sub-basis of the corresponding
topology consists of set of the form {F ∈ F(G) : F ∩ U 6= ∅} where U varies over the open subsets
of G. First we show that
R = {(F, a) ∈ F(G) × I : F ⊆ f −1 (La )}
is a closed set. Let (F, a) ∈ F(G) × I, (Fn , an ) ∈ R (n < ω) be such that limn<ω (Fn , an ) = (F, a).
Let x ∈ F be arbitrary; then by Fn ⊆ f −1 (Lan ) (n < ω) there are xn ∈ f −1 (Lan ) (n < ω) such that
limn<ω xn = x. Since f is continuous, f (x) = limn<ω f (xn ). By ≤⊆ I × I being closed, f (xn ) ≤ an
(n < ω) implies f (x) ≤ a. Thus x ∈ f −1 (La ). Since x ∈ F was arbitrary, F ⊆ f −1 (La ) thus
(F, a) ∈ R follows.
Observe that {(K, F ) ∈ K(2ω ) × F(G) : K ⊆ cl2ω (F )} is a Borel set (see e.g. the proof of [11,
Theorem 12.6 p. 75]). Hence
S = {(K, F, a) ∈ K(2ω ) × F(G) × I : K ⊆ cl2ω (F ), F ⊆ f −1 (La )}
is also Borel. It is obvious that
T = {(K, a) ∈ K(2ω ) × I : K ⊆ cl2ω (f −1 (La ))} = PrK(2ω )×I (S),
so T is an analytic set. Since I is analytic, I = PrK(2ω ) (T ) is analytic, as well.
We show that g : I → I, g(a) = cl2ω (f −1 (La )) (a ∈ I) is a Tukey map. Let A ⊆ I be arbitrary.
If g[A] ⊆ I is bounded then there is a b ∈ I satisfying cl2ω (f −1 (La )) ⊆ cl2ω (f −1 (Lb )) (a ∈ A).
Then f −1 (a) ⊆ cl2ω (f −1 (Lb )) ∩ G = f −1 (Lb ) (a ∈ A), so a ≤ b (a ∈ A). Thus g maps unbounded
sets into unbounded sets, as stated.
Finally let h : I → I be any function satisfying K ⊆ cl2ω (f −1 (Lh(K) )) (K ∈ I); we show that h
is a Tukey map. Let A ⊆ I be arbitrary. If h[A] ⊆ I is bounded then there is a b ∈ I satisfying
h(K) ≤ b hence Lh(K) ⊆ Lb (K ∈ A). Then K ⊆ cl2ω (f −1 (Lb )) (K ∈ A); i.e. h maps unbounded
sets into unbounded sets, as required.
Next we present a simple proof of Proposition 5.5.
Proof of Proposition 5.5. Observe that for every n < ω, [2ω ]≤n ⊆ K(2ω ) is closed. Hence [2ω ]<ω ⊆
K(2ω ) is an Fσ ideal of compact sets. Then, with the notation of Definition 5.17, by Lemma 5.18.5
and Lemma 5.19, Imax = (Φ[[2ω ]<ω ])↓ fulfills the requirements.
28
TAMÁS MÁTRAI
As we mentioned above, Imax is not basic. To see this, let us recall the definition of basic directed
partial orders. Recall that a set is called bounded if it is bounded from above.
Definition 5.27. ([24, Definition p. 1881]) Let (D, ≤) be a separable metric space endowed with
a partial order. Then (D, ≤) is basic if
(1) each pair of elements of D has the least upper bound with respect to ≤ and the binary
operation of least upper bound from D × D to D is continuous;
(2) each bounded sequence has a converging subsequence;
(3) each converging sequence has a bounded subsequence.
Since every infinite subset of Imax is unbounded, in a topology making Imax satisfy Definition
5.27.3 there are no injective convergent sequences. The only such metric topology is the discrete
topology, which is not separable in this case.
We close this section with the proof of Proposition 5.4. The construction originates from [16,
Theorem 6 p. 183], and an analogous construction was used in [17] to show that the structure
(P(ω), ⊆? ) embeds into the family of Fσ ideals on ω partially ordered by Borel reduction. Our main
improvement compared to [16] and [17], which makes possible to omit definability assumptions, is
that our proof for non-reducibility is purely combinatorial.
We define sequences (bj )j<ω and (mj )j<ω by induction on j, as follows. Set m0 = 0; if j < ω
and mj is already defined, set bj = 2j·mj and mj+1 = mj + bj .
Let Ij = [mj , mj+1
stand for logarithm of base 2. For every S ∈ [ω]ω and
S) (j < ω). Let log S
N < ω, let IS (N ) = j∈S∩N Ij and IS = j∈S Ij . For every j < ω and x ⊆ ω,
kxkj =
log(|x ∩ Ij | + 1)
, kxk = sup kxkj .
mj + 1
j<ω
We define FS = {x ⊆ IS : supj<ω kxkj < ∞}. For every N < ω, let FS (N ) = {x ∈ FS : kxk ≤ N }.
We will use the property that for arbitrary n, j < ω and (xi )i<n ⊆ P(ω),
[ log(n)
.
(5.4)
xi ≤ sup kxi kj +
m
i<n
j +1
i<n
j
Proposition 5.4 is an immediate corollary of the following two statements.
Proposition 5.28. For every S ∈ [ω]ω , FS ⊆ P(ω) is an Fσ ideal which is not basic in any
topology.
Proposition 5.29. For every S, T ∈ [ω]ω , FS ≤T FT if and only if S ⊆? T .
Proof of Proposition 5.28. It is obvious that FS (N ) ⊆ P(ω) (N < ω) are closed sets, so FS is
an Fσ ideal. Suppose there is a topology on FS which makes it basic. For every N < ω, let
(xi (N ))i<ω ⊆ [ω]<ω be a sequence such that
(1) for every i < ω we have xi (N ) ⊆ IS , N − 1 ≤ kxi (N )k ≤ N ;
(2) for every j < ω we have |{i < ω : xi (N ) ∩ Ij 6= ∅}| ≤ 1.
S
Set Xi (N ) = i≤k<ω xk (N ) (i, N < ω). Then by (2), Xi (N ) ⊆ IS and kXi (N )k ≤ N , i.e.
Xi (N ) ∈ FS (i, N < ω).
Fix N < ω. The sequence (xk (N ))k<ω is bounded by X0 (N ), so by Definition 5.27.2, it has a
convergent subsequence (xk (N ))k∈IN for some IN ∈ [ω]ω .
INFINITE DIMENSIONAL PERFECT SET THEOREMS
29
By [24, Lemma 3.1 p. 1882], for every i < ω, Xi+ (N ) = {y ∈ FS : y ⊆ Xi (N )} is compact. Since the convergent
sequence (xk (N ))k∈IN is eventually a subsequence of Xi+ (N ) (i < ω),
T
+
limk∈IN xk (N ) ∈ i<ω Xi (N ) = {∅}. Thus limi∈IN xi (N ) = ∅ (N < ω).
Since a basic topology is metric, by an easy diagonalization argument we can find a function
ϕ : ω → ω such that ϕ(N ) ∈ IN (N < ω) and limN <ω xϕ(N ) (N ) = ∅. Thus (xϕ(N ) (N ))N <ω is
a convergent sequence, which by (1) has no bounded subsequence. This contradicts Definition
5.27.3.
The non-reduction part of Proposition 5.29 is based on the following. For every t ∈ 2≤ω we set
[t = 1] = {n < ω : t(n) = 1}. Recall Ns = {x ∈ P(ω) : x ∩ |s| = [s = 1]} (s ∈ 2<ω ).
Lemma 5.30. Let S, T ∈ [ω]ω satisfy S ∩ T = ∅. Let A ⊆ FS (1), n ∈ S and s ∈ 2mn be such
that Ns ∩ FS (1) 6= ∅ and A is of second category everywhere in Ns ∩ FS (1) in the relative topology
of FS (1). Let N < ω, and let f : A → FT (N ) be an arbitrary function. Then there is a t ∈ 2mn ,
such
S that [t = 1] ⊆ T , ktk ≤ N , B = {x ∈ A : f (x)|mn = t} ⊆ FS (1) is of second category and
k Bkn ≥ n − 2.
Proof. Let T = {t ∈ 2mn : [t = 1] ⊆ T, ktk ≤ N }. For every t ∈ T , set S
Bt = {x ∈ A : f (x)|mn = t}.
If there is a t ∈ T such that Bt is of second category in FS (1) and k Bt kn ≥ n − 2 then we are
done.
So suppose no such t ∈ T exists. Set
C=
[
{Bt : t ∈ T , Bt is of second category in FS (1)}.
Then by (5.4), using |T | ≤ 2mn ,
log(2mn )
log(2mn )
<n−2+
< n − 1.
mn + 1
mn + 1
S
S
However,
since A is of second category everywhere in Ns ∩ FS (1), we get In ⊆ A = C. Thus
S
k Ckn = log(2n·mn + 1)/(mn + 1) ≥ n − 1. This contradiction completes the proof.
k
S
Ckn ≤ supt∈T k
S
Bt k +
Proof of Proposition 5.29. Let first S, T ∈ [ω]ω satisfy S ⊆? T . Set f : FS → FT , f (x) = x ∩ IT
(x ∈ FS ). Is is easy to verify that f is a Tukey map.
Let now S, T ∈ [ω]ω satisfy S 6⊆? T . As we have seen, FS\T ≤T FS . So it is enough to prove
FS\T 6≤T FT ; that is, we can assume S ∩ T = ∅.
Let f : FS → FT be an arbitrary function. We find N < ω and ni < ω, si , ti ∈ 2<ω , Ai ⊆ FS (1)
(i < ω) such that
(1)
(2)
(3)
(4)
ni ∈ S, ni < ni+1 , si ∈ 2mni (i < ω);
ti ∈ 2mni , kti k ≤ N , ti v ti+1 , [ti = 1] ⊆ T (i < ω);
Nsi ∩ FS (1) 6= ∅ and Ai is of second
S category everywhere in Nsi ∩ FS (1) (i < ω);
for every i < w, f [Ai ] ⊆ FT (N ), k Ai+1 kni ≥ ni − 2 and f (x)|mni = ti (x ∈ Ai+1 ).
Suppose first the construction is done;
S we show that fSis not a Tukey map. For every i < ω, let
Xi+1 ⊆ A
be
minimal
such
that
(
Xi+1 ) ∩ Ini = ( S
Ai+1 ) ∩ Ini . Then |Xi | ≤ bni (i < ω). By
i+1
S
S
(4), x = i<ω Xi satisfies kxkni ≥ ni − 2 (i < ω), i.e. i<ω Xi is unbounded in FS .
30
TAMÁS MÁTRAI
S
S
We show that i<ω {f (x) : x ∈ Xi } ⊆ FT is bounded. Set τ = i<ω ti ; then by (2), τ ∈ FT and
kτ k ≤ N . For every n ∈ T , by (5.4), (4) and n ∈
/ S,
[ [
[
{f (x) : x ∈ Xi } ≤ τ ∪ {f (x) : x ∈ Xi , ni < n}
n
i<ω
n
P
log(mn + 2)
log(1 + {bni : ni < n} + 1)
≤N+
≤ N + 1,
≤N+
mn + 1
mn + 1
so the statement follows.
S
It remains to perform the construction. We have f [FS (1)] ⊆ N <ω FT (N ) so there are N < ω
and A0 ⊆ FS (1) such that f [A0 ] ⊆ FT (N ) and A0 is of second category in FS (1). So we can find
n0 ∈ S, s0 ∈ 2n0 such that Ns0 ∩ FS (1) 6= ∅ and A0 is of second category everywhere in Ns0 ∩ FS (1).
Let i < ω be arbitrary and suppose that ni , si , and Ai have already been found. We apply
Lemma 5.30 with S, T , N , A = Ai , n = ni and s = si . We get a ti ∈ 2mni such
S that [ti = 1] ⊆ T ,
kti k ≤ N , Ai+1 = {x ∈ Ai : f (x)|mni = ti } ⊆ FS (1) is of second category and k Ai+1 kni ≥ ni − 2.
It remains to choose ni+1 ∈ S, ni+1 > ni , si+1 ∈ 2ni+1 such that Nsi+1 ∩ FS (1) 6= ∅ and Ai+1 is of
second category everywhere in Nsi+1 ∩ FS (1). Then (1), (3) and (4) hold, and for (2) it remains
to show ti−1 v ti if i > 0. However, this follows from Ai+1 ⊆ Ai using the inductive hypothesis
f (x)|mni−1 = ti−1 (x ∈ Ai ). So the proof is complete.
6. Problems
In our present work we did not apply some well-understood methods for studying infinite dimensional perfect set theorems. We close this paper with a survey of possible further research
directions, and state some related open problems.
6.1. General symmetric topologies, analytic relations. Apart from the usual product topology, there are several topologies on Rω which are symmetric, i.e. open sets remain open under
arbitrary permutation of coordinates, and which are important in applications; e.g. the box topology or the topologies induced by the `p (1 ≤ p ≤ ∞) norms. There is no reason to believe that
the product topology is the most appropriate for the formulation of optimal infinite dimensional
perfect set theorems. It would be interesting to find weaker largeness assumptions in these finer
topologies than our X ω \ A ∈ M in Theorem 4.1, which still imply the existence of a non-empty
perfect A-homogeneous set. E.g. it is easy to construct a symmetric dense open set A ⊆ Rω such
that there is no infinite A-homogeneous set; but in Rω endowed with the box topology, for every
symmetric dense open set there is an infinite homogeneous set. Compare this with the remarks
following Corollary 2.3.
Note also that ironically, all of our perfect set theorems hold for co-analytic relations, while
the rank approach of [13] and [22] is able to handle analytic relations. Observe that to every
construction using finite approximations, one can associate the tree of finite approximations ordered
by end-extension, such that the ill-foundedness of this tree is equivalent with the existence of the
limit object of the construction. Therefore the infinite dimensional counterpart of [13, Proposition
4.1 p. 151] is natural to formulate and easy to prove. The more involved task would be to study the
existence of universal relations as in [13, Section 5], and to characterize the resulting rank as in [22].
It seems that these investigations can be carried out for any of the above mentioned refinements of
the product topology, as well.
INFINITE DIMENSIONAL PERFECT SET THEOREMS
31
6.2. More fusion games. It seems informative to study fusions of other games of descriptive set
theory, especially those of Separation games and Wadge games. We remark that the way we increased the dimension in our fusion scheme was completely arbitrary; different schemes characterize
other notions of smallness.
We propose an explicit modification of Gω of Definition 3.2 which seems particularly interesting to
study. Consider the game Gωm ⊆ Gω , where in addition,
in the (n + 1)th move player I is required to
S
play (Ui (2(n + 1 − i)))i≤n+1 such that Un+1 (0) ⊆ i≤n Ui (2(n − i) + 1); else the game is unchanged.
For player II this game is easier to win, still one can show that the existence of a winning strategy
for player II in Gωm (A) implies the existence of a non-empty perfect A-homogeneous set.
Unfortunately we could not characterize the existence of a winning strategy for player I in Gωm (A).
Nevertheless, we expect that fusion games modified in such ways can provide sharper results.
6.3. Other Choquet topologies and forcings. The Ellentuck topology and the density topologies are Choquet, in particular Theorem 4.1 can be applied to them. However, they fail the
Kuratowski-Ulam property, which is crucial for Lemma 4.4 and so for all the results of Section
4.1 and Section 4.2. E.g. our methods does not allow us to prove the counterparts of Theorem 1.2
and Theorem 4.10, in which “meager” is replaced by “Lebesgue null” and “Cohen” is replaced by
“random”.
Problem 6.1. Let A ⊆ Rω be a co-analytic set. Does the existence of an A-homogeneous set of
positive outer Lebesgue measure imply the existence of a non-empty perfect set A-homogeneous set?
Problem 6.2. Let V be a model obtained from a model of the Continuum Hypothesis by adding ω2
random reals. In V , let A ⊆ Rω be a co-analytic set. Does the existence of an A-homogeneous set
of cardinality ω2 imply the existence of a non-empty perfect set A-homogeneous set?
Note that the finite dimensional analogue of Problem 6.1 holds for every Lebesgue measurable
set A by the measure version of Mycielski’s Theorem (see e.g. [27, Theorem 6.40]), while the finite
dimensional analogue of Problem 6.2 holds for every analytic set A by [22, Fact 1.16 p. 23].
These problems may be related to the following.
Problem 6.3. Let A ⊆ 2ω ×2ω be a co-meager set. Is it true in ZFC that there is an A-homogeneous
set which is of second category everywhere?
If e.g. cof(M) = cov(M), then by an easy transfinite argument, for every co-meager set A ⊆
2ω × 2ω one can construct an A-homogeneous set which is of second category everywhere. Also note
the counterpart of Problem 6.3 involving Lebesgue measure fails.
Theorem 6.4. ([2], [7]) Let V be a model obtained from a model of the Continuum Hypothesis by
adding ω2 Cohen reals. Then in V , there is an Fσ set A ⊆ 2ω × 2ω of Lebesgue measure 1 such that
there exists no A-homogeneous set of positive outer Lebesgue measure.
The proof of Theorem 6.4 is based on the observation that in Cohen extensions, if for an Fσ
set A ⊆ 2ω × 2ω there is an A-homogeneous set of positive outer Lebesgue measure then there
is an A-homogeneous set of positive Lebesgue measure. Thus it is likely that Problem 6.1 has
an affirmative answer in Cohen extensions. However, the proof of Theorem 6.4 does not have a
straightforward modification valid for random extensions, so we could not obtain a negative answer
to Problem 6.3.
Problem 6.3 was motivated by the question whether the largeness assumption in Theorem 1.2 is
the natural analogue of the largeness assumption of Mycielski’s Theorem. As we pointed out above,
32
TAMÁS MÁTRAI
this is consistently true, since for sets A ⊆ 2ω × 2ω having the Baire property, being co-meager and
having an A-homogeneous set which is of second category everywhere are consistently equivalent.
In Cohen extensions there are perfectly meager sets of cardinality 2ℵ0 , so the largeness assumption
in Theorem 1.2 is consistently not optimal. It would be interesting to know whether the converse
is also consistent; the following is also related to Problem 6.3.
Problem 6.5. Is it consistent with ZFC that H ⊆ R is perfectly meager if and only if there is an
open set A ⊆ Rω such that ISω (H) ⊆ A but there is no non-empty perfect A-homogeneous set?
6.4. Local results. In [15] the following “local” infinite dimensional perfect set theorem was obtained (see also [27, Corollary 6.48]). We call a sequence (xn )n<ω ∈ Rω rapidly increasing if
0 < xn+2 − xn+1 < xn+1 − xn (n < ω). The set of rapidly increasing sequences is denoted by R.
Theorem 6.6. ([15, Theorem 2 p. 275]) For every finite Baire measurable coloring of R there is a
perfect set P ⊆ R such that P ω ∩ R is monochromatic.
Ramsey-type theorems like Theorem 6.6 are very important, e.g. Theorem 5.14 is also based on
Theorem 6.6. The most general results of this nature are known as the Halpern-Läuchli Theorems
(see e.g. [10] or [27] and the references therein). It is easy to see that there is an open tower U
such that [U] = R, in particular R ∈
/ M. It would be very useful to explore the possible interplay
between our approach and such local results.
6.5. Cofinal types of analytic ideals. A deeper analysis of the construction in the proof of
Theorem 5.21 reveals that it uses that E of Definition 5.23 is an equivalence relation in an essential
way. Thus this method cannot yield a Borel ideal satisfying the conditions of Theorem 5.21. On
the other hand, the construction in the proof of Theorem 2.1 is the base step of the construction
of an Fσ ideal satisfying the conditions of Theorem 5.21. Therefore we expect a positive answer to
the following problems.
Problem 6.7. Is there an Fσ ideal I ⊆ P(ω) such that [ω1 ]<ω ≤T I but I has no non-empty
perfect strongly unbounded subsets?
Problem 6.8. ([16, Conjecture 1 p. 194]) Is there an Fσ ideal I ⊆ P(ω) such that [ω1 ]<ω ≤T I
but ω ω 6≤T I?
We also expect the consistency of the failure of the primality property for Imax .
Problem 6.9. Is it consistent with ZFC that Imax fails the primality property, i.e. there exist
analytic ideals I, J ⊆ P(ω) such that Imax ≤T I ⊕ J but Imax 6≤T I and Imax 6≤T J ?
At the present stage of research, one could wonder whether for every analytic ideal I ⊆ P(ω),
[ω2 ]<ω ≤T I implies that I has a non-empty perfect strongly unbounded subset. However, we
expect that the affirmative answer to Problem 6.7 will be based on a construction which is flexible
enough to rule out such speculations.
Note also that presently the iterated perfect set model is our only example where the conclusion
of Theorem 5.7.1 holds. Nevertheless, we think that a better understanding of infinite dimensional
perfect set theorem vs. Lebesgue measure and random extensions will provide more models where
such results hold. Especially because by [19], in the random real model every universal measure
zero set has cardinality ≤ ω1 .
INFINITE DIMENSIONAL PERFECT SET THEOREMS
33
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Department of Mathematics, University of Toronto, 40 St. George St., Toronto, ON, Canada
E-mail address: matrait@renyi.hu