On uniformly continuous functions
between pseudometric spaces and
the Axiom of Countable Choice
Samuel G. da Silva
UFBA, Salvador/Bahia/Brazil
ICM 2018
Short Communication
Dedicated to the memory of Prof. Horst Herrlich (1937-2015)
Rio de Janeiro, RJ, Brazil – August 02, 2018
Samuel G. da Silva
ICM 2018 Short Communications
The research on weak choice principles
In this talk, we work within ZF – i.e., Zermelo-Fraenkel choiceless set
theory – and the main goal of this line of research is to investigate how
certain statements (from Topology ? from Analysis ? from Algebra ? from Set Theory ? from Order
Theory ?
) are positioned in the hierarchy of weak choice principles.
Our main choice principle, for this presentation, is ACω – the
Axiom of Countable Choice, which declares that every countable
family of non-empty sets has a choice function.
ACω is a key ingredient of many classical, ZFC results on metric
and pseudometric spaces (and we assume the audience is familiar
with the usual terminology concerning such topological spaces).
Essentially, a pseudometric space is a generalized metric space in which the distance between two distinct points
is allowed to be zero.
Samuel G. da Silva
ICM 2018 Short Communications
ACω and pseudometric spaces
The following results are due to Bentley and Herrlich:
Theorem (Bentley and Herrlich, 1998)
ACω is enough to prove, in the realm of pseudometric spaces, the
equivalence between the notions of:
(i) separability;
(ii) having a countable base; and
(iii) being a Lindelöf space.
Moreover, ACω is, in fact, equivalent to each of the statements
“All pseudometric spaces with a countable base are separable”
and “All Lindelöf pseudometric spaces are separable” .
A similar phenomenon happens if one considers the statements “Sequentially compact pseudometric spaces are
totally bounded” , “Totally bounded, complete pseudometric spaces are compact” and “Sequentially compact,
pseudometric spaces are compact”(all mentioned results are also due to Bentley and Herrlich).
Samuel G. da Silva
ICM 2018 Short Communications
ACω and UC spaces
In several recent papers, Keremedis has investigated (under the
lenses of the weak choice principles theory) the class of metric
spaces which are Atsuji, or UC – i.e., the class of metric spaces
satisfying the following property: all continuous real valued
functions defined on them are, necessarily, uniformly
continuous.
Among many other results, the theorem of the following slide was
established by Keremedis; always recall that all theorems in this
talk are theorems of ZF.
Samuel G. da Silva
ICM 2018 Short Communications
ACω and UC spaces
Theorem (Keremedis, 2016)
Let M = (M, d) be a metric space. Under ACω , the following
statements are equivalent:
(i) M is Lebesgue – i.e., every open cover of M has a Lebesgue
covering number.
(ii) M is UC.
(iii) M is uniformly normal – i.e., the distance between any two
disjoint, non-empty closed subsets of M is strictly positive.
It is worthwhile remarking that, in the previous theorem,
implications (i) ⇒ (ii) ⇒ (iii) hold in ZF.
Samuel G. da Silva
ICM 2018 Short Communications
Our results: investigating the property UC
for pseudometric spaces
In this work we show that, if we consider the property of being UC
in the class of pseudometric spaces, we get that the statement
“All sequentially compact, pseudometric spaces are UC ”
is, itself, an equivalent of the Axiom of Countable Choice.
We also show that ACω is equivalent to a well-known
characterization (via pairs of sequences) of uniform continuity of
real valued continuous functions defined on pseudometric spaces.
Samuel G. da Silva
ICM 2018 Short Communications
The main theorem
The following theorem includes all results of this work.
Theorem (da S., 2018 - to appear in Archive for Mathematical Logic) TFAE:
(i) ACω .
(ii) If M, N are pseudometric spaces and f : M → N is a continuous function, f is
uniformly continuous if, and only if, for every pair of (not necessarily convergent)
sequences (xn ), (yn ) ∈ M such that dM (xn , yn ) → 0 one has dN (f (xn ), f (yn )) →
0.
(iii) If M is a pseudometric space and f : M → R is a continuous function, f is
uniformly continuous if, and only if, for every pair of (not necessarily convergent)
sequences (xn ), (yn ) ∈ M such that dM (xn , yn ) → 0 one has |f (xn ) − f (yn )| → 0.
(iv ) Let M, N be pseudometric spaces and assume M to be sequentially compact.
Then, for every function f : M → N, f is continuous if, and only if, it is uniformly
continuous.
(v ) Sequentially compact, pseudometric spaces are UC.
Samuel G. da Silva
ICM 2018 Short Communications
The display of implications
In the preceding theorem, (ii) ⇒ (iii) and (iv ) ⇒ (v ) are obvious.
Proofs of (i) ⇒ (ii) ⇒ (iv ) are apparent and go as expected
(formalizing within ZF the “naive” proofs one could be found in
any general topology textbook; one has just to point out, precisely,
in which parts of the argument the choice principle ACω is used).
9A (i)
(iii)
ks
(ii)
ks
(v )
+3
KS
(iv )
One is able to check in the display that (ii) is the crucial statement in our context – at least in the etymological
sense of the word “crucial” (which refers to: of the form of a cross; cross-shaped).
Samuel G. da Silva
ICM 2018 Short Communications
The display of implications
What is left to prove is (iii) ⇒ (i) and (v ) ⇒ (i), and the proof of
both implications will be made at once.
9A (i)
(iii) ks
Samuel G. da Silva
(ii)
ks
(v )
+3
KS
(iv )
ICM 2018 Short Communications
A contrapositive argument
As
(iii) If M is a pseudometric space and f : M → R is a continuous function, f is
uniformly continuous if, and only if, for every pair of (not necessarily
convergent) sequences (xn ), (yn ) ∈ M such that dM (xn , yn ) → 0 one has
|f (xn ) − f (yn )| → 0;
and
(v ) Sequentially compact, pseudometric spaces are UC,
it follows that (iii) ⇒ (i) and (v ) ⇒ (i) can be proved at once if we show that:
(*) Under the failure of ACω , there is sequentially compact pseudometric space
M with a continuous, non-uniformly continuous real valued function f : M → R
satisfying, for all pairs of sequences (xn ), (yn ),
dM (xn , yn ) → 0 =⇒ |f (xn ) − f (yn )| → 0.
Samuel G. da Silva
ICM 2018 Short Communications
The road to (*): characterizing the failure of ACω
The following statement is “folklore”.
Under the failure of the Axiom of Countable Choice, there is a
countable family F = {Xn : n > 1} of non-empty sets such that
no infinite subfamily has a choice function.
Indeed, if the statement
(†) “All countable families of non-empty sets have infinite subfamilies
which have choice functions.”
given any countable
holds, then we get ACω through the following argument:Q
family {An : n > 1} of non-empty sets , an element g ∈
An is easily
n>1
produced, after applying (†) to the countable
Q family of non-empty sets
Ai for all n > 2; indeed, if f is an
{Xn : n > 1} given by: X1 = A1 and Xn =
i6n
infinite partial choice function for the Xn ’s, then let g (n) be the nth component
of f (k), where k is the first natural number > n in the domain of f .
Samuel G. da Silva
ICM 2018 Short Communications
The road to (*): enlarging the points of a metric space
to blobs of diameter zero
The importance of working with pseudometric spaces rather than
metric spaces comes from the following observation: a
pseudometric may allow you to “enlarge” the points of a metric
space to “blobs” of diameter zero.
Theorem (Bentley and Herrlich, 1998)
Under the failure of the Axiom of Countable Choice, there is a noncompact, non-separable, sequentially compact pseudometric space.
Samuel G. da Silva
ICM 2018 Short Communications
The road to (*): enlarging the points of a metric space
to blobs of diameter zero
Indeed: suppose ACω fails. Let F = {Xn : n > 1} be as in the “folklore
statement” we have
S just mentioned: we may assume w.l.g. that F is a disjoint
family. Let M =
Xn , and define d : M × M → R as follows: for any
n>1
x, y ∈ X ,
d(x, y ) =
0
| n1 −
1
|
m
if there is m > 1 such that x, y ∈ Xm ; and
if n 6= m, x ∈ Xn and y ∈ Xm .
M is sequentially compact since, for any sequence (yn )n>1 of points of M,
necessarily the set A = {m > 1 : Xm contains points of the sequence (yn )n>1 }
is finite (otherwise the infinite subfamily of F given by {Xm : m ∈ A} would
have – clearly – a choice function). It follows that (yn )n>1 has at least one
subsequence (ynk )k>1 satisfying d(ynk , ynj ) = 0 for all k, j > 1 and any such
subsequence converges.
A similar reasoning shows that M is not separable.
To see that M is not compact, notice that the open cover given by
{Xn : n > 1} has no finite subcover.
Samuel G. da Silva
ICM 2018 Short Communications
The proof of (*)
(*) Under the failure of ACω , there is sequentially compact pseudometric space
M with a continuous, non-uniformly continuous real valued function f : M → R
satisfying, for all pairs of sequences (xn ), (yn ),
dM (xn , yn ) → 0 =⇒ |f (xn ) − f (yn )| → 0.
As expected, let M be the sequentially compact pseudometric
space constructed in the previous slide.
We define a function f : M → R in the following way: for every
x ∈ X,
f (x) = n ⇐⇒ x ∈ Xn .
Samuel G. da Silva
ICM 2018 Short Communications
The proof of (*)
If we think of each point of the discrete set A = { n1 : n > 1} as being a “blob of
diameter zero” – such that all points of Xm (which have distance zero to each
other, by definition) are “compressed” all together into the point m1 , then the
graph of f looks exactly like the graph of the function f (x) = x1 defined on A.
f is continuous (since it is locally constant) and a straigthforward calculation
shows that it is not uniformly continuous.
Let (xn ), (yn ) be sequences in M such that dM (xn , yn ) → 0. Let Ax and Ay
denote the finite subsets of N given by
Ax = {m > 1 : Xm ∩ im(x) 6= ∅} , and Ay = {m > 1 : Xm ∩ im(y ) 6= ∅}.
If dM (xn , yn ) → 0, then necessarily there is some m ∈ Ax ∩ Ay and some n0 > 1
such that, for every n > n0 , both points xn and yn are elements of Xm . It
follows that f (xn ) = f (yn ) = m for all n > n0 , and so we are done.
Samuel G. da Silva
ICM 2018 Short Communications
References
Bentley, H. L., Herrlich, H., Countable choice and pseudometric spaces.
8th Prague Topological Symposium on General Topology and Its Relations
to Modern Analysis and Algebra (1996). Topology and its Applications
85, 1-3 (1998), 153–164.
Herrlich, H., Axiom of choice, volume 1876 of Lecture Notes in
Mathematics, Springer-Verlag, Berlin, xiv + 194 pp. (2006)
Keremedis, K., On metric spaces where continuous real valued functions
are uniformly continuous in ZF. Topology and its Applications 210
(2016), 366–375.
Keremedis, K., Uniform continuity and normality of metric spaces in ZF.
Bulletin of the Polish Academy Sciences. Mathematics 65 (2017), 113–124
Howard, P., Rubin, J. E., Consequences of the axiom of choice, volume 59
of Mathematical Surveys and Monographs, American Mathematical
Society, Providence, RI, viii + 432 pp. (1998)
Samuel G. da Silva
ICM 2018 Short Communications
Thanks and I hope see you soon in Salvador !
Samuel G. da Silva
ICM 2018 Short Communications
A model of ZF where ACω fails
The model usually denoted by M1 and known in the literature as
Cohen’s basic model, is obtained after adding (through a
symmetric forcing extension) a countable set of generic reals.
In this model,
There is an infinite subset of reals which has no countable
subset. One may assume such set is dense. (Cohen)
As a subspace of R, the preceding subset gives a secound
countable, non-separable space.
The discrete space N is separable and has a countable base,
but fails to be Lindelöf. (Herrlich/Strecker)
Each one of the preceding items ensure that ACω fails in the
Cohen’s basic model !
Samuel G. da Silva
ICM 2018 Short Communications
How the “expected proofs” go . . .
First, two ZF facts on continuous functions defined between
pseudometric spaces:
Fact 1
If M, N are pseudometric spaces and f : M → N is continuous at a
point a ∈ M, then d(f (a), f (b)) = 0 whenever d(a, b) = 0.
Indeed, if d(a, b) = 0 and f is continuous at a then d(f (a), f (b)) < ε for every
ε > 0.
Fact 2
If M, N are pseudometric spaces and f : M → N is continuous at
a point x0 ∈ M, then f (xn ) → f (x0 ) (in N) whenever (xn ) is a
sequence of points of M with xn → x0 (in M).
In fact, what needs Countable Choice to be established is the converse
statement.
Samuel G. da Silva
ICM 2018 Short Communications
Recall the statements . . .
Theorem (da S.) TFAE:
(i) ACω .
(ii) If M, N are pseudometric spaces and f : M → N is a continuous function, f is
uniformly continuous if, and only if, for every pair of (not necessarily convergent)
sequences (xn ), (yn ) ∈ M such that dM (xn , yn ) → 0 one has dN (f (xn ), f (yn )) →
0.
(iii) If M is a pseudometric space and f : M → R is a continuous function, f is
uniformly continuous if, and only if, for every pair of (not necessarily convergent)
sequences (xn ), (yn ) ∈ M such that dM (xn , yn ) → 0 one has |f (xn ) − f (yn )| → 0.
(iv ) Let M, N be pseudometric spaces and assume M to be sequentially compact.
Then, for every function f : M → N, f is continuous if, and only if, it is uniformly
continuous.
(v ) Sequentially compact, pseudometric spaces are UC.
Samuel G. da Silva
ICM 2018 Short Communications
Proof of (i) ⇒ (ii)
(ii) is a “if, and only if ” statement. The “only if ” part is obvious.
For the “if ” part, we argue contrapositively: assume f is not
uniformly continuous. So, there is some ε > 0 such that for all
δ > 0 there are points x, y ∈ M with dM (x, y ) < δ and
dN (f (x), f (y )) > ε.
This gives us a family {An : n > 1} of non-empty subsets of
M × M such that, for any n > 1,
An = {(x, y ) ∈ M × M : dM (x, y ) <
1
and dN (f (x), f (y )) > ε}.
n
Let zn = (xn , yn ) ∈ An be given by Countable Choice; we have that
dM (xn , yn ) → 0 whereas dN (f (xn ), f (yn )) 6→ 0, as desired.
Samuel G. da Silva
ICM 2018 Short Communications
Proof of (ii) ⇒ (iv )
Again, the “if ” part is obvious.
For the “only if ” part, let M, N be pseudometric spaces, with M
sequentially compact, and let f : M → N be a continuous function.
Assuming (ii), what we have to check is that
dN (f (xn ), f (yn )) → 0
whenever (xn ), (yn ) are sequences in M satisfying dM (xn , yn ) → 0.
So, let (xn ), (yn ) ∈ M satisfy dM (xn , yn ) → 0. Assume towards a
contradiction that dN (f (xn ), f (yn )) 6→ 0; without loss of generality,
we may assume there is some ε > 0 such that dN (f (xn ), f (yn )) > ε
for all n > 1.
Samuel G. da Silva
ICM 2018 Short Communications
Proof of (ii) ⇒ (iv )
By sequential compactness, we may also assume, without loss of
generality, the sequences (xn ), (yn ) to be convergent – and
therefore there are points x, y ∈ M such that xn → x and yn → y ;
as dM (xn , yn ) → 0 we have, necessarily, dM (x, y ) = 0.
The continuity of f ensures that d(f (x), f (y )) = 0, by Fact 1.
However, the continuity of f also ensures f (xn ) → f (x) and
f (yn ) → f (y ) in N, by Fact 2.
Then, necessarily one has dN (f (xn ), f (yn )) → 0, but this is an
absurdity since dN (f (xn ), f (yn )) > ε for all n > 1. It follows that
the desired implication holds.
Samuel G. da Silva
ICM 2018 Short Communications