Houston Journal of Mathematics
c 2006 University of Houston
­
Volume 32, No. 1, 2006
A CLASS OF ONE–DIMENSIONAL, NONLOCALLY
CONNECTED CONTINUA FOR WHICH THE SET FUNCTION
T IS CONTINUOUS
SERGIO MACÍAS
Communicated by Charles Hagopian
Abstract. We present a class of one–dimensional, nonlocally connected
continua for which the set function T is continuous.
1. Introduction
F. Burton Jones defined the set function T in [6]. Since then many properties
related with this function have been studied.
In 1970, David Bellamy asked in [1] if any nonlocally connected continuum for
which T is continuous had to be indecomposable. In 1983, Bellamy (see Remark
2 in page 10 of [2]) answered the question in the negative showing that the circle
of pseudo–arcs [4] is a decomposable nonlocally connected continuum for which
T is continuous.
In 1985, Wayne Lewis constructed [7], for each one–dimensional continuum X,
b such that X
b has a terminal continuous decoma one–dimensional continuum X
position into pseudo–arcs such that the decomposition space is homeomorphic to
X.
In section 3 we show that if X is a locally connected one–dimensional continb constructed by
uum, then the set function T is continuous for the continuum X
Lewis.
2000 Mathematics Subject Classification. 54C60; 54B20.
Key words and phrases. Aposyndetic continuum, circle of pseudo–arcs, continuous decomposition, continuum, decomposable continuum, hyperspace, indecomposable continuum, n–
aposyndeict continuum, n–fold symmetric product, point T –symmetric continuum, pseudo–arc,
set function T , T –additive continuum.
161
162
SERGIO MACÍAS
2. Definitions
If (Z, d) is a metric space, then given A ⊂ Z the interior of A is denoted by
A .
A continuum is a compact connected metric space. A subcontinuum of a space
Z is a continuum contained in Z. A continuum is decomposable if it is the union
of two of its proper subcontinua. A continuum is indecomposable if it is not
decomposable. A subcontinuum Y of a continuum X is said to be terminal
provided that for any subcontinuum K of X such that K ∩ Y 6= ∅, we have that
either K ⊂ Y or Y ⊂ K.
Given a continuum X, a decomposition of X is a family G of nonempty and
S
mutually disjoint subsets of X such that G = X. A decomposition G of a
continuum X is said to be continuous provided that the quotient map q : X →
X/G is open and closed.
The pseudo–arc is the only nondegenerate hereditarily indecomposable chainable continuum. The circle of pseudo–arcs is a one–dimensional circularly chainable plane continuum admitting a continuous decomposition into pseudo–arcs
such that the decomposition space is the unit circle S 1 .
Given a continuum X, we define the set function T as follows: if A ⊂ X then
◦
T (A) = X \ {x ∈ X | there exists a subcontinuum W of X such that
x ∈ W ◦ ⊂ W ⊂ X \ A}.
We write TX if there is a possibility of confusion. Let us observe that for any
subset A of X, T (A) is closed subset of X and A ⊂ T (A). A continuum X is
said to be point T –symmetric if for each pair, p and q, of points of X, p 6∈ T ({q})
if and only if q 6∈ T ({p}). A continuum X is said to be T –additive provided that
for each pair of closed subsets A and B of X, T (A ∪ B) = T (A) ∪ T (B). Given a
positive integer n, a continuum X is n–aposyndetic provided that for each subset
A of X having n points, T (A) = A. If the continuum X is 1–aposyndetic, then
we say that X is aposyndetic.
We say that T is continuous for a continuum X provided that T : 2X → 2X
is continuous, where 2X is the hyperspace of nonempty closed subsets of X,
topologized with the Hausdorff metric (see [9]).
Given a positive integer n and a continuum X , we define the n–fold symmetric
product of X, denoted Fn (X), by
Fn (X) = {A ∈ 2X | A has at most n points}.
THE SET FUNCTION T IS CONTINUOUS
163
3. Theorems
We begin this section recalling a theorem by Bellamy, a proof of this result
may be found in Theorem 5, page 9, of [2]:
3.1. Theorem. Let X and Z be continua, where Z is locally connected. If f : X →
Z is a surjective, monotone and open map such that for each proper subcontinuum
W of X, f (W ) 6= Z, then TX is continuous for X.
Next, we state the theorem by Lewis, a proof of this result may be found in
[7]:
3.2. Theorem. If X is a one–dimensional continuum, then there exists a one–
b such that X
b has a terminal continuous decomposition
dimensional continuum X
into pseudo–arcs such that the decomposition space is homeomorphic to X.
3.3. Remark. Observe that the continua constructed by Wayne Lewis do not
contain arcs. Hence, they are not locally connected and, by construction, the
b is decomposable if and only if X is decomposable.
continuum X
Now, we are ready to prove our main result:
3.4. Theorem. If X is a locally connected one–dimensional continuum, then the
b given in Theorem 3.2.
set function TXb is continuous for the continuum X
b be
Proof. Let X be a one–dimensional locally connected continuum, and let X
b
the one–dimensional continuum given in Theorem 3.2. Since X admits a continb → X is monotone,
uous decomposition into pseudo–arcs, the quotient map q : X
b let x̂ ∈ X
b \ W,
open and surjective. Let W be a proper subcontinuum of X,
and let Px̂ the pseudo–arc of the decomposition that contains x̂. Since Px̂ is a
b either W ⊂ Px̂ or W ∩ Px̂ = ∅. In any case, q(W ) is
terminal subcontinuum of X,
a proper subcontinuum of X. Therefore, by Theorem 3.1, TXb is continuous. £
3.5. Remark. What Bellamy did to show the continuity of TX in Theorem 3.1 was
b of Theorem 3.4,
to prove that TX (A) = f −1 ◦ f (A). Thus, for the continuum X
b is point T –symmetric. Hence, X
b is T –
TXb ({x̂}) = Px̂ . This implies that X
additive by Lemma 9 of [1].
The following example shows that the requirement in Theorem 3.1, that the
images of proper subcontinua of the domain are proper subcontinua of the range
e is due to Karen Villarreal [10]:
is essential. The existence of the continuum Z
3.6. Example. Let Z be the circle of pseudo–arcs. Then there exists a two–
e ⊂ Z×Z
dimensional aposyndetic nonlocally connected homogeneous continuum Z
164
SERGIO MACÍAS
e→
such that if π1 is the projection map of Z ×Z onto the first factor, then π1 |Ze : Z
1
Z is open, monotone and surjective. Let q̃ : Z → S be the quotient map from
e → S 1 is open, monotone and surjective.
Z onto the unit circle. Then q̃ ◦ π1 |Ze : Z
e that ∆Z = {(z, z) | z ∈ Z} is a
It is easy to see, from the construction of Z,
¡
e
proper subcontinuum of Z such that q̃ ◦ π1 |Ze (∆Z ) = S 1 . Observe that TZe is
e is aposyndetic, we
not continuous. To see this, suppose it is continuous. Since Z
e
e
have that Z is point T –symmetric. Hence, Z is T –additive by Lemma 9 of [1].
e T e (A) = S{T e ({a}) | a ∈
By Theorem B of [3], for any closed subset A of Z
Z
Z
S
e is locally connected (see page 5 of [1]), a
A} = {{a} | a ∈ A} = A. Thus, Z
contradiction. Therefore, TZe is not continuous.
Next, we show that if T is continuous on a product of two continua, then each
factor must be locally connected. To this end, let us recall a result proved by
FitzGerald (see Corollary 2.1 of [5]):
3.7. Theorem. Let X and Y be nondegenerate continua. If K is a countable
closed subset of X × Y , then T (K) = K. In particular, X × Y is n–aposyndetic
for each positive integer n.
As a consequence of this result we have:
3.8. Corollary. Let Y and Z be nondegenerate continua. If X = Y × Z is a
continuum for which T is continuous, then X is locally connected. In particular,
Y and Z are locally connected.
Proof. Let A be a nonempty closed subset of X. Since A is compact, there
exists a sequence {Kn }∞
n=1 of finite subsets of X converging to A. Since T is
continuous, {T (Kn )}∞
=1 converges to T (A). By Theorem 3.7, T (Kn ) = Kn for
each positive integer n. Hence, T (A) = A. Since A was an arbitrary nonemtpy
closed subset of X, X is locally connected (see page 5 of [1]).
£
The next result is the analog to FitzGerald’s Theorem for symmetric products.
A proof of it may be found in Theorem 8 of [8]:
3.9. Theorem. If X is a continuum, n ≥ 2 and K is any closed, countable subset
of Fn (X), then TFn (X) (K) = K, that is, Fn (X) is countable closed aposyndetic.
In particular, Fn (X) is m–point aposyndetic for each positive integer m.
As a consequence of Theorem 3.9, we have the following:
3.10. Corollary. Let X be a continuum and let n ≥ 2. If TFn (X) is continuous
for Fn (X), then X is locally connected.
THE SET FUNCTION T IS CONTINUOUS
165
Proof. As in the proof of Corollary 3.8, TFn (X) (A) = A for each closed subset
A of Fn (X). Hence, Fn (X) is locally connected (see page 5 of [1]). Therefore, X
is locally connected by Lemma 2 of [8].
£
We end the paper with the following question:
3.11. Question. Let X be a nonlocally connected decomposable continuum for
which the set function T is continuous. Does X admit a terminal continuous
decomposition into pseudo–arcs such that the decomposition space is locally connected?
The author thanks the referee for the suggestions made which improved the
paper.
REFERENCES
[1] D. P. Bellamy, Continua for Which the Set Function T is Continuous, Trans. Amer. Math.
Soc., 151 (1970) 581–587.
[2] D. P. Bellamy, Some Topics in Modern Continua Theory, in Continua Decompositions
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(1983), 1–26.
[3] D. P. Bellamy and H. S. Davis, Continuum Neighborhoods and Filterbases, Proc. Amer.
Math. Soc., 27 (1971), 371–374.
[4] R. H. Bing and F. B. Jones, Another Homogeneous Plane Continuum, Trans. Amer. Math.
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[5] R. W. FitzGerald, The Cartesian Product of Non–degenerate Compact Continua is n–
point Aposyndetic, in Topology Conference (Arizona State Univ., Tempe, Ariz., 1967),
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[6] F. B. Jones, Concerning Nonaposyndetic Continua, Amer. Math., 70 (1948), 403–413.
[7] W. Lewis, Continuous Curves of Pseudo–arcs, Houston J. Math., 11 (1985), 91–99.
[8] S. Macı́as, Aposyndetic Properties of Symmetric Products of Continua, Topology Proc.,
22 (1997), 281–296.
[9] S. B. Nadler, Jr., Hyperspaces of Sets, Monographs and Textbooks in Pure and Applied
Math., Vol. 49, Marcel Dekker, New York, Basel, 1978.
[10] K. Villarreal, Fibered Products of Homogeneous Continua, Trans. Amer. Math. Soc., 338
(1993), 933–939.
Received December 8, 2003
Revised version received July 22, 2004
(Instituto de Matemáticas, U. N. A. M., Circuito Exterior, Ciudad Universitaria, México,
D. F., C. P. 04510. MEXICO)
E-mail address: macias@servidor.unam.mx