A strategy for finding CCC and non-separable spaces

In this post we present a general strategy for finding CCC spaces that are not separable. As illustration, we give four implementations of this strategy.

In searching for counterexamples in topology, one good place to look is of course the book by Steen and Seebach [2]. There are four examples of spaces that are CCC but not separable found in [2] – counterexamples 20, 21, 24 and 63. Counterexamples 20 and 21 are not Hausdorff. Counterexample 24 is the uncountable Fort space (it is completely normal but not perfectly normal). Counterexample 63 (Countable Complement Extension Topology) is Hausdorff but is not regular. These are valuable examples especially the last two (24 and 63). The examples discussed below expand the offerings in Steen and Seebach.

The discussion of CCC but not separable in this post does not use axioms beyond the usual axioms of set theory (i.e. ZFC). The discussion here does not touch on Suslin lines or other examples that require extra set theory. The existence of Suslin lines is independent of ZFC. A Suslin line would produce an example of a perfectly normal first countable CCC non-separable space. In models of set theory where Suslin lines do not exist, a perfectly normal first countable CCC non-separable space can also be produced using other set-theoretic assumptions. The examples discussed below are not as nice as the set-theoretic examples since they usually are not first countable and perfectly normal.

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The countable chain conditon

A topological space X is said to have the countable chain condition (to have the CCC for short) if \mathcal{U} is a disjoint collection of non-empty open subsets of X (meaning that for any A,B \in \mathcal{U} with A \ne B, we have A \cap B=\varnothing), then \mathcal{U} is countable. In other words, in a space with the CCC, there cannot be uncountably many pairwise disjoint non-empty open sets. For ease of discussion, if X has the CCC, we also say that X is a CCC space or X is CCC. A space X is separable if there exists a countable subset A of X such that A is dense in X (meaning that if U is a nonempty open subset of X, then U \cap A \ne \varnothing).

It is clear that any separable space has the CCC. In metric spaces, these two properties are equivalent. Among topological spaces in general, the two properties are not identical. Thus “CCC but not separable” is one way to distinguish between metrizable spaces and non-metrizable spaces. Even in non-metrizable spaces, “CCC but not separable” is also a way to obtain more information about the spaces being investigated.

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The strategy

Here’s the strategy for finding CCC and not separable.

    The strategy is to narrow the focus to spaces where “CCC and not separable” is likely to exist. Specifically, look for a space or a class of spaces such that each space in the class has the countable chain condition but is not hereditarily separable. If the non-separable subspace is also a dense subspace of the starting space, it would be “CCC and not separable.”

Any dense subspace of a CCC space always has the CCC. Thus the search focuses on the subspaces in a CCC space that are reliably CCC. The strategy is to find non-separable spaces among these dense subspaces. The search is given an assist if the space or class of spaces in question has a characteristic that delineate the “separable” from the CCC (see Example 3 and Example 4 below).

In the following sections, we illustrate four different ways to apply the strategy.

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Example 1

The first way is a standard example found in the literature. The space to start from is the product space of separable spaces, which is always CCC. By a theorem of Ross and Stone, the product of more than continuum many separable spaces is not separable. Thus one way to get an example of CCC but not separable space is to take the product of more than continuum many separable spaces. For example, if c is the cardinality of continuum, then consider \left\{0,1 \right\}^{2^c}, the product of 2^c many copies of \left\{0,1 \right\}, or consider X^{2^c} where X is your favorite separable space.

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Example 2

The second implementation of the strategy is also from taking the product of separable spaces. This time the number of factors does not have to be more than continuum. Here, we focus on one particular dense subspace of the product space, the \Sigma-products. To make this clear, let’s focus on a specific example. Consider X=\left\{0,1 \right\}^{c} where c is the cardinality of continuum. Consider the following subspace.

    \Sigma(\left\{0,1 \right\}^{c})= \left\{x \in X: x(\alpha) \ne 0 \text{ for at most countably many } \alpha < c \right\}

The subspace \Sigma(\left\{0,1 \right\}^{c}) is dense in X, thus has CCC. It is straightforward to verify that \Sigma(\left\{0,1 \right\}^{c}) is not separable.

To implement this example, find any space X which is a product space of separable spaces, each of which has at least two point (one of the points is labeled 0). The dense subspace is the \Sigma-product, which is the subspace consisting of all points that are non-zero at only countably many coordinates. The \Sigma-product has the countable chain condition since it is a dense subspace of the CCC space X. The \Sigma-product is not separable since there are uncountably many factors in the product space X and that each factor has at least two points. This idea had been implemented in this previous post.

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Example 3

The third class of spaces is the class of Pixley-Roy spaces, which are hyperspaces. Given a space X, let \mathcal{F}[X] be the set of all non-empty finite subsets of X. For F \in \mathcal{F}[X] and for any open subset U of X, let [F,U]=\left\{B \in \mathcal{F}[X]: F \subset B \subset U \right\}. The sets [F,U] over all F and U form a base for a topology on \mathcal{F}[X]. This topology is called the Pixley-Roy topology (or Pixley-Roy hyperspace topology). The set \mathcal{F}[X] with this topology is called a Pixley-Roy space.

The Pixley-Roy hyperspaces are discussed in this previous post. Whenever the ground space X is uncountable, \mathcal{F}[X] is not a separable space. We need to identify the \mathcal{F}[X] that are CCC. According to the previous post on Pixley-Roy hyperspaces, for any space X with a countable network, \mathcal{F}[X] is CCC. Thus for any uncountable space X with a countable network, the Pixley-Roy space \mathcal{F}[X] is a CCC space that is not separable. The following gives a few such examples.

    \mathcal{F}[\mathbb{R}]

    \mathcal{F}[X] where X is any uncountable, separable and metrizable space.

    \mathcal{F}[X] where X is uncountable and is the continuous image of a separable metrizable space.

Spaces with countable networks are discussed in this previous post. An example of a space X that is the continuous image of a separable metrizable space is the bow-tie space found this previous post. Another example is any quote space of a separable metrizable space.

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Example 4

For the fourth implementation of the strategy, we go back to the product space of separable spaces in Example 2, with the exception that the focus is on the product of the real line \mathbb{R}. Let X be any uncountable completely regular space. The product space \mathbb{R}^X always has the CCC since it is a product of separable space. Now we single out a dense subspace of \mathbb{R}^X for which there is a characterization for separability, namely the subspace C(X), which is the set of all continuous functions from X into \mathbb{R}. The subspace C(X) as a topological space is usually denoted by C_p(X). For a basic discussion of C_p(X), see this previous post.

We know precisely when C_p(X) is separable. The following theorem captures the idea.

Theorem 1 – Theorem I.1.3 [1]
The function space C_p(X) is separable if and only if the domain space X has a weaker (or coarser) separable metric topology (in other words, X is submetrizable with a separable metric topology).

Based on the theorem, C_p(X) is separable for any separable metric space X. Other examples of separable C_p(X) include spaces X that are created by tweaking the usual Euclidean topology on the real line and at the same time retaining the usual real line topology as a weaker topology, e.g. the Sorgenfrey line and the Michael line. Thus C_p(X) would be separable if X is a space such as the Sorgenfrey line or the Michael line. For our purpose at hand, we need to look for spaces that are not like the Sorgenfrey line or the Michael line. Here’s some examples of spaces X that have no weaker separable metric topology.

  • Any compact space X that is not metrizable.
  • The space X=\omega_1, the first uncountable ordinal with the order topology.
  • Any space X=C_p(Y) where Y is not separable.

The function space C_p(X) for any one of the above three spaces has the CCC but is not separable. It is well known that any compact space with a weaker metrizable topology is metrizable. Some examples for compact X are: the first uncountable successor ordinal \omega_1+1, the double arrow space, and the product space \left\{0,1 \right\}^{\omega_1}.

It can be shown that C_p(\omega_1) is not separable (see this previous post). The last example is due to the following theorem.

Theorem 2 – Theorem I.1.4 [1]
The function space C_p(Y) has a weaker (or coarser) separable metric topology if and only if the domain space Y is separable.

Thus picking a non-separable space Y would guarantee that C_p(Y) has a weaker separable metric topology. As a result, C_p(C_p(Y)) is a CCC and not separable space.

Interestingly, Theorem 1 and Theorem 2 show a duality existing between the property of having a weaker separable metric topology and the property of being separable. The two theorems allow us to switch the two properties between the domain space and the function space.

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Remarks

Another interesting point to make is that Theorem 1 and Theorem 2 together allow us to “buy one get one free.” Once we obtain a space X that is CCC and not separable from any one of the avenues discussed here, the function space C_p(X) has no weaker separable metric topology (by Theorem 2) and the function space C_p(C_p(X)) is another example of CCC and not separable.

The strategy discussed above unifies all four examples. Undoubtedly there will be other examples that can come from the strategy. To find more examples, find a space or a class of spaces that are reliably CCC and then look for potential non-separable spaces among the dense subspaces of the starting space.

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Exercises

  1. Show that in metrizable spaces, CCC and separable are equivalent. The only part that needs to be shown is that if X is metrizable and CCC, then X is separable.
  2. Show that any dense subspace of a CCC space is also CCC.
  3. Verify that the space \Sigma(\left\{0,1 \right\}^{c}) defined in Example 2 is dense in X and is not separable.
  4. Verify that the Pixley-Roy space \mathcal{F}[\mathbb{R}] defined in Example 3 is CCC and not separable.
  5. Verify that function space C_p(\omega_1) mentioned in Example 4 is not separable. Hint: use the pressing down lemma.

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Reference

  1. Arkhangelskii, A. V., Topological Function Spaces, Mathematics and Its Applications Series, Kluwer Academic Publishers, Dordrecht, 1992.
  2. Steen, L. A., Seebach, J. A., Counterexamples in Topology, Dover Publications, Inc., New York, 1995.

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\copyright \ 2016 \text{ by Dan Ma}

Normal dense subspaces of a product of “continuum” many separable metric factors

Is every normal dense subspace of a product of separable metric spaces collectionwise normal? This question was posed by Arkhangelskii in [1] (see Problem I.5.25). A partial positive answer is provided by a theorem that is usually attributed to Corson: If Y is a normal dense subspace of a product of separable metric spaces and if Y \times Y is also normal, then Y is collectionwise normal. In this post, using a simple combinatorial argument, we show that any normal dense subspace of a product of continuum many separable metric space is collectionwise normal (see Corollary 4 below), which is a corollary of the following theorem.

Theorem 1
Let X be a normal space with character \le 2^\omega. If 2^\omega<2^{\omega_1}, then the following holds:

  • If Y is a closed and discrete subspace of X with \lvert Y \lvert=\omega_1, then Y contains a separated subset of cardinality \omega_1.

Theorem 1 gives the corollary indicated at the beginning and several other interesting results. The statement 2^\omega<2^{\omega_1} means that the cardinality of the power set (the set of all subsets) of \omega is strictly less than the cardinality of the power set of \omega_1. Note that the statement 2^\omega<2^{\omega_1} follows from the continuum hypothesis (CH), the statement that 2^\omega=\omega_1. With the assumption 2^\omega<2^{\omega_1}, Theorem 1 is a theorem that goes beyond ZFC. We also present an alternative to Theorem 1 that removes the assumption 2^\omega<2^{\omega_1} (see Theorem 6 below).

A subset T of a space S is a separated set (in S) if for each t \in T, there is an open subset O_t of S with t \in O_t such that \left\{O_t: t \in T \right\} is a pairwise disjoint collection. First we prove Theorem 1 and then discuss the corollaries.

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Proof of Theorem 1

Suppose Y is a closed and discrete subset of X with \lvert Y \lvert=\omega_1 such that no subset of Y of cardinality \omega_1 can be separated. We then show that 2^{\omega_1} \le 2^{\omega}.

For each y \in Y, let \mathcal{B}_y be a local base at the point y such that \lvert \mathcal{B}_y \lvert \le 2^\omega. Let \mathcal{B}=\bigcup_{y \in Y} \mathcal{B}_y. Thus \lvert \mathcal{B} \lvert \le 2^\omega. By normality, for each W \subset Y, let U_W be an open subset of X such that W \subset U_W and \overline{U_W} \cap (Y-W)=\varnothing. For each W \subset Y, consider the following collection of open sets:

    \mathcal{G}_W=\left\{V \in \mathcal{B}_y: y \in W \text{ and } V \subset U_W  \right\}

For each W \subset Y, choose a maximal disjoint collection \mathcal{M}_W of open sets in \mathcal{G}_W. Because no subset of Y of cardinality \omega_1 can be separated, each \mathcal{M}_W is countable. If W_1 \ne W_2, then \mathcal{M}_{W_1} \ne \mathcal{M}_{W_2}.

Let \mathcal{P}(Y) be the power set (i.e. the set of all subsets) of Y. Let \mathcal{P}_\omega(\mathcal{B}) be the set of all countable subsets of \mathcal{B}. Then the mapping W \mapsto \mathcal{M}_W is a one-to-one map from \mathcal{P}(Y) into \mathcal{P}_\omega(\mathcal{B}). Note that \lvert \mathcal{P}(Y) \lvert=2^{\omega_1}. Also note that since \lvert \mathcal{B} \lvert \le 2^\omega, \lvert \mathcal{P}_\omega(\mathcal{B}) \lvert \le 2^\omega. Thus 2^{\omega_1} \le 2^{\omega}. \blacksquare

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Some Corollaries of Theorem 1

Here’s some corollaries that follow easily from Theorem 1. A space X has the countable chain condition (CCC) if every pairwise disjoint collection of non-empty open subset of X is countable. For convenience, if X has the CCC, we say X is CCC. The following corollaries make use of the fact that any normal space with countable extent is collectionwise normal (see Theorem 2 in this previous post).

Corollary 2
Let X be a CCC space with character \le 2^\omega. If 2^\omega<2^{\omega_1}, then the following conditions hold:

  • If X is normal, then every closed and discrete subset of X is countable, i.e., X has countable extent.
  • If X is normal, then X is collectionwise normal.

Corollary 3
Let X be a CCC space with character \le 2^\omega. If CH holds, then the following conditions hold:

  • If X is normal, then every closed and discrete subset of X is countable, i.e., X has countable extent.
  • If X is normal, then X is collectionwise normal.

Corollary 4
Let X=\prod_{\alpha<2^\omega} X_\alpha be a product where each factor X_\alpha is a separable metric space. If 2^\omega<2^{\omega_1}, then the following conditions hold:

  • If Y is a normal dense subspace of X, then Y has countable extent.
  • If Y is a normal dense subspace of X, then Y is collectionwise normal.

Corollary 4 is the result indicated in the title of the post. The product of separable spaces has the CCC. Thus the product space X and any dense subspace of X have the CCC. Because X is a product of continuum many separable metric spaces, X and any subspace of X have characters \le 2^\omega. Then Corollary 4 follows from Corollary 2.

When dealing with the topic of normal versus collectionwise normal, it is hard to avoid the connection with the normal Moore space conjecture. Theorem 1 gives the result of F. B. Jones from 1937 (see [3]). We have the following theorem.

Theorem 5
If 2^\omega<2^{\omega_1}, then every separable normal Moore space is metrizable.

Though this was not how Jones proved it in [3], Theorem 5 is a corollary of Corollary 2. By Corollary 2, any separable normal Moore space is collectionwise normal. It is well known that collectionwise normal Moore space is metrizable (Bing’s metrization theorem, see Theorem 5.4.1 in [2]).

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A ZFC Theorem

We now prove a result that is similar to Corollary 2 but uses no set-theory beyond the Zermelo–Fraenkel set theory plus axiom of choice (abbreviated by ZFC). Of course the conclusion is not as strong. Even though the assumption 2^\omega<2^{\omega_1} is removed in Theorem 6, note the similarity between the proof of Theorem 1 and the proof of Theorem 6.

Theorem 6
Let X be a CCC space with character \le 2^\omega. Then the following conditions hold:

  • If X is normal, then every closed and discrete subset of X has cardinality less than continuum.

Proof of Theorem 6
Let X be a normal CCC space with character \le 2^\omega. Let Y be a closed and discrete subset of X. We show that \lvert Y \lvert < 2^\omega. Suppose that \lvert Y \lvert = 2^\omega.

For each y \in Y, let \mathcal{B}_y be a local base at the point y such that \lvert \mathcal{B}_y \lvert \le 2^\omega. Let \mathcal{B}=\bigcup_{y \in Y} \mathcal{B}_y. Thus \lvert \mathcal{B} \lvert = 2^\omega. By normality, for each W \subset Y, let U_W be an open subset of X such that W \subset U_W and \overline{U_W} \cap (Y-W)=\varnothing. For each W \subset Y, consider the following collection of open sets:

    \mathcal{G}_W=\left\{V \in \mathcal{B}_y: y \in W \text{ and } V \subset U_W  \right\}

For each W \subset Y, choose \mathcal{M}_W \subset \mathcal{G}_W such that \mathcal{M}_W is a maximal disjoint collection. Since X is CCC, \mathcal{M}_W is countable. It is clear that if W_1 \ne W_2, then \mathcal{M}_{W_1} \ne \mathcal{M}_{W_2}.

Let \mathcal{P}(Y) be the power set (i.e. the set of all subsets) of Y. Let \mathcal{P}_\omega(\mathcal{B}) be the set of all countable subsets of \mathcal{B}. Then the mapping W \mapsto \mathcal{M}_W is a one-to-one map from \mathcal{P}(Y) into \mathcal{P}_\omega(\mathcal{B}). Note that since \lvert \mathcal{B} \lvert = 2^\omega, \lvert \mathcal{P}_\omega(\mathcal{B}) \lvert = 2^\omega. Thus \lvert \mathcal{P}(Y) \lvert \le 2^{\omega}. However, Y is assumed to be of cardinality continuum. Then \lvert \mathcal{P}(Y) \lvert>2^{\omega_1}, leading to a contradiction. Thus it must be the case that \lvert Y \lvert < 2^\omega. \blacksquare

With Theorem 6, Corollary 3 still holds. Theorem 6 removes the set-theoretic assumption of 2^\omega<2^{\omega_1}. As a result, the upper bound for cardinalities of closed and discrete sets is (at least potentially) higher.

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Reference

  1. Arkhangelskii, A. V., Topological Function Spaces, Mathematics and Its Applications Series, Kluwer Academic Publishers, Dordrecht, 1992.
  2. Engelking, R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.
  3. Jones, F. B., Concerning normal and completely normal spaces, Bull. Amer. Math. Soc., 43, 671-677, 1937.

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\copyright \ 2014 \text{ by Dan Ma}

Working with the function space Cp(X)

This post provides basic information about the space of real-valued continuous functions with the pointwise convergence topology. The goal is to discuss the setting and to define the standard basic open sets in the function space, providing background information for subsequent posts.

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Completely Regular Spaces

The starting point is a completely regular space. A space X is said to be completely regular if X is a T_0 space and for each x \in X and for each closed subset A of X with x \notin A, there is a continuous function f:X \rightarrow [0,1] such that f(A) \subset \left\{0 \right\} and f(x)=1. Note that the T_0 axiom and the existence of the continuous function imply the T_1 axiom, which is equivalent to the property that single points are closed sets. Completely regular spaces are also called Tychonoff spaces.

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Defining the Function Space C_p(X)

Let X be a completely regular space. Let C(X) be the set of all real-valued continuous functions defined on the space X. The set C(X) is naturally a subspace of the product space \prod_{x \in X} \mathbb{R}=\mathbb{R}^X. Thus C(X) can be endowed with the subspace topology inherited from the product space \mathbb{R}^X. When this is the case, the function space C(X) is denoted by C_p(X). The topology on C_p(X) is called the pointwise convergence topology.

Now we need a good handle on the open sets in the function space C_p(X). A basic open set in the product space \mathbb{R}^X is of the form \prod_{x \in X} U_x where each U_x is an open subset of \mathbb{R} such that U_x = \mathbb{R} for all but finitely many x \in X (equivalently U_x \ne \mathbb{R} for only finitely many x \in X). Thus a basic open set in C_p(X) is of the form:

    C(X) \cap \biggl(\prod_{x \in X} U_x \biggr) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)

where each U_x is an open subset of \mathbb{R} and U_x = \mathbb{R} for all but finitely many x \in X. In addition, when U_x \ne \mathbb{R}, we can take U_x to be an open interval of the form (a,b). To simplify notation, the basic open sets as described in (1) can also be notated by:

    \prod_{x \in X} U_x \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1a)

Thus when working with open sets in C_p(X), we take \prod_{x \in X} U_x to mean the set of all f \in C(X) such that f(x) \in U_x for each x \in X.

To make the basic open sets of C_p(X) more explicit, (1) or (1a) is translated as follows:

    \bigcap_{x \in F} \ [x, O_x] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)

where F \subset X is a finite set, for each x \in F, O_x is an open interval of \mathbb{R}, and [x,O_x] is the set of all f \in C(X) such that f(x) \in O_x.

There is another description of basic open sets that is useful. Let f \in C_p(X). Let F \subset X be finite. Let \epsilon>0. Let B(f,F,\epsilon) be defined as follows:

    B(f,F,\epsilon)=\left\{g \in C(X): \forall \ x \in F, \lvert f(x)-g(x) \lvert< \epsilon \right\} \ \ \ \ \ \ \ (3)

In proving results about C_p(X), we can use basic open sets that are described in any one of the three forms (1), (2) and (3). If U is a basic open subset of C_p(X), as described in (1) or (1a), we use supp(U) to denote the finite set of x \in X such that U_x \ne \mathbb{R}. The set supp(U) is called the support of U. The support for the basic open sets as described in (2) and (3) is already explicitly stated.

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Basic Discussion

The theory of C_p(X) is a vast subject area. For a systematic introduction, see [1]. One fundamental theme in function space theory is the study of how properties of X and C_p(X) are related. The domain space X and the function space C(X) are not on the same footing. The domain X only has a topological structure. The function space C_p(X) carries a topology and two natural algebraic operations of addition and multiplication, making it a topological ring. In addition, C_p(X) can be regarded as a topological group, or a linear topological space. In this post and in many subsequent posts, we narrow the focus to the topological properties of X and C_p(X), paying attention to the how the topological properties of X and C_p(X) are related.

In addition to the pointwise convergence topology, there are other topologies that can be defined on C(X), e.g., the compact-open topology, the topology of uniform convergence and others. Both the pointwise convergence topology and the compact-open topology are examples of set-open topologies. In this post and in many of the subsequent posts, the focus is on the pointwise convergence topology, i.e., the subspace topology on C(X) inherited from the product space.

The space C_p(X) automatically inherits certain properties of the product space \mathbb{R}^X. Note that C(X) is dense in \mathbb{R}^X. The product \mathbb{R}^X has the countable chain condition (CCC) since it is a product of separable spaces. Hence C_p(X) always has the CCC, i.e., there are no uncountably many pairwise disjoint open subsets of C_p(X), regardless what the domain space X is. One consequence of the CCC is that C_p(X) is paracompact if and only if C_p(X) is Lindelof.

It is well known that \mathbb{R}^X is separable if and only if the cardinality of X \le continuum. Since C(X) is dense in \mathbb{R}^X, C_p(X) is not separable if the cardinality of X > continuum. Thus C_p(X) is one way to get a CCC space that is not separable. There are non-separable C_p(X) where the cardinality of X \le continuum. Obtaining such C_p(X) would require more than the properties of the product space \mathbb{R}^X; using properties of X would be necessary.

The properties of C_p(X) discussed so far are inherited from the product space. Refer to chapter one of [1] for other elementary properties of C_p(X). See this post for a discussion of C_p(X) where X is a separable metric space. See this post about a consequence of normality of C_p(X).

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Reference

  1. Arkhangelskii, A. V., Topological Function Spaces, Mathematics and Its Applications Series, Kluwer Academic Publishers, Dordrecht, 1992.

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\copyright \ 2014 \text{ by Dan Ma}

A factorization theorem for products of separable spaces

Let X=\prod_{\alpha \in A} X_\alpha be a product space. Let f:X \rightarrow T be a continuous function where T is a topological space. In this post, we discuss what it means for the continuous function f to depend on countably many coordinates and then discuss some conditions that we can impose on the product space and on the range space T to ensure that every continuous f defined on the product space will depend on countably many coordinates. This notion of a continuous function depending on countably many coordinates is equivalent to factoring the continuous function into the composition of a projection map and a continuous function defined on a countable subproduct (see Lemma 1 below).

Let’s set some notation about the product space we work with in this post. Let X=\prod_{\alpha \in A} X_\alpha be a product space. Let T be a topological space. Let f:X \rightarrow T be continuous. For any B \subset A, \pi_B is the natural projection from the full product space X=\prod_{\alpha \in A} X_\alpha into the subproduct \prod_{\alpha \in B} X_\alpha. Standard basic open sets of X=\prod_{\alpha \in A} X_\alpha are of the form \prod_{\alpha \in A} O_\alpha where each O_\alpha is open in X_\alpha and that O_\alpha=X_\alpha for all but finitely many \alpha \in A. We use supp(\prod_{\alpha \in A} O_\alpha) to denote the finite set of \alpha \in A where O_\alpha \ne X_\alpha.

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Factoring a Continuous Map

The function f is said to depend on countably many coordinates if there exists a countable set B \subset A such that for any x,y \in X, if x_\alpha=y_\alpha for all \alpha \in B, then f(x)=f(y). Suppose f is instead defined on a subspace Y of X. The function f is said to depend on countably many coordinates if there exists a countable B \subset A such that for any x,y \in Y, if x_\alpha=y_\alpha for all \alpha \in B, then f(x)=f(y).

We have the following lemmas.

Lemma 1

    Let X=\prod_{\alpha \in A} X_\alpha be a product space. Let T be a topological space. Let f:X \rightarrow T be continuous. Then the following are equivalent.

    1. There exists a countable B \subset A such that for any x,y \in X, if x_\alpha=y_\alpha for all \alpha \in B, then f(x)=f(y).
    2. There exists a countable B \subset A such that f=g \circ \pi_B where g: \prod_{\alpha \in B} X_\alpha \rightarrow T is continuous.

Lemma 1a

    Let X=\prod_{\alpha \in A} X_\alpha be a product space. Let T be a topological space. Let Y be a dense subspace of X. Let f:Y \rightarrow T be continuous. Then the following are equivalent.

    1. There exists a countable B \subset A such that for any x,y \in Y, if x_\alpha=y_\alpha for all \alpha \in B, then f(x)=f(y).
    2. There exists a countable B \subset A such that f=g \circ \pi_B where g: \pi_B(Y) \rightarrow T is continuous.

It is straightforward to verify Lemma 1 and Lemma 1a. We use condition 1 to define what it means for a function to be dependent on countably many coordinates. Both lemmas indicate that either condition is a valid definition. These two lemmas also indicate why the notion being discussed can be called a factorization notion.

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When a Continuous Map Can Be Factored

We discuss some conditions that we can place on the product space X=\prod_{\alpha \in A} X_\alpha and on the range space T so that any continuous map depends on countably many coordinates. We prove the following theorem.

Theorem 1

    Let X=\prod_{\alpha \in A} X_\alpha be a product space such that each factor X_\alpha is a separable space. Let T be a second countable space (i.e. having a countable base). Then for any dense subspace Y of X, any continuous function f:Y \rightarrow T depends on countably many coordinates, i.e., either one of the conditions in Lemma 1a holds.

Before stating the main theorem, we need one more lemma. Let W \subset X=\prod_{\alpha \in A} X_\alpha. The set W is said to depend on countably many coordinates if there exists a countable B \subset A such that for any x \in W and for any y \in X, if x_\alpha=y_\alpha for all \alpha \in B, then y \in W.

When we try to determine whether a function f:Y \rightarrow T, where Y \subset X, can be factored, we will need to decide whether a set W \subset Y depends on countably many coordinates. Let W \subset Y \subset X=\prod_{\alpha \in A} X_\alpha. The set W is said to depend on countably many coordinates if there exists a countable B \subset A such that for any x \in W and for any y \in Y, if x_\alpha=y_\alpha for all \alpha \in B, then y \in W. We have the following lemma.

Lemma 2

    Let X=\prod_{\alpha \in A} X_\alpha be a product space with the countable chain condition. Let Y be a dense subspace of X.

    1. Let U be an open subset of X. Then \overline{U} depends on countably many coordinates.
    2. Let W be an open subset of Y. Then \overline{W} depends on countably many coordinates (closure in Y).

Proof of Lemma 2
Proof of Part 1
Let U \subset X be open. Let \mathcal{B} be a collection of pairwise disjoint open subsets of the open set U \subset X such that \mathcal{B} is maximal with this property, i.e., if you throw one more open set into \mathcal{B}, it will be no longer pairwise disjoint. Let V=\bigcup \mathcal{B}. Since \mathcal{B} is maximal, \overline{V}=\overline{U}. Since X has the countable chain condition, \mathcal{B} is countable.

Let B=\bigcup \left\{supp(O): O \in \mathcal{B} \right\}. The set B is a countable subset of A since B is the union of countably many finite sets. We have the following claims.

Claim 1
The open set V depends on the coordinates in B.

Let x \in V and y \in X such that x_\alpha=y_\alpha for all \alpha \in B. We need to show that y \in V. Firstly, x \in O for some O \in \mathcal{B}. It follows that x_\alpha=y_\alpha for all \alpha \in supp(O). Thus y \in O \subset V. This completes the proof of Claim 1.

Claim 2
The set \overline{V} depends on the coordinates in B.

Let x \in \overline{V} and y \in X such that x_\alpha=y_\alpha for all \alpha \in B. We need to show y \in \overline{V}. To this end, let O=\prod_{\alpha \in A} O_\alpha be a standard basic open set with y \in O. The goal is to find some q \in O \cap V. Define G=\prod_{\alpha \in A} G_\alpha such that G_\alpha=O_\alpha for all \alpha \in B and G_\alpha=X_\alpha for all \alpha \in A-B. Then x \in G. Since x \in \overline{V}, there exists some p \in V \cap G. Define q such that q_\alpha=p_\alpha for all \alpha \in B and q_\alpha=y_\alpha for all \alpha \in A-B. Since supp(V) \subset B, q \in V. On the other hand, q \in O. This completes the proof of Claim 2.

As noted above, \overline{V}=\overline{U}. Thus \overline{U} depends on countably many coordinates, namely the coordinates in the set B. This completes the proof of Part 1.

Proof of Part 2
For any S \subset X, let \overline{S} denote the closure of S in Y. Let Cl_X(S) denote the closure of S in X. Let W \subset Y be open. Let W_1 be open in X such that W=W_1 \cap Y. By Part 1, Cl_X(W_1) depends on countably many coordinates, say the coordinates in the countable set B \subset A. This means that for any x \in Cl_X(W_1) and for any y \in X, if x_\alpha=y_\alpha for all \alpha \in B, then y \in Cl_X(W_1). Thus for any x \in \overline{W} and for any y \in Y, if x_\alpha=y_\alpha for all \alpha \in B, then y \in Cl_X(W_1). If we have y \in \overline{W}, then we are done. So we only need to show that if y \in Y and y \in Cl_X(W_1), then y \in \overline{W}. This is why we need to assume Y is dense in X.

Let y \in Y and y \in Cl_X(W_1). Let C be an open subset of Y with y \in C. There exists an open subset D of X such that C=D \cap Y. Then D \cap Cl_X(W_1) \ne \varnothing. Note that D \cap W_1 is open and D \cap W_1 \ne \varnothing. Since Y is dense in X, D \cap W_1 must contain points of Y. These points of Y are also points of W. Thus C contains points of W. It follows that y \in \overline{W}. This concludes the proof of Part 2. \blacksquare

Proof of Theorem 1
Let Y be a dense subspace of X=\prod_{\alpha \in A} X_\alpha. Let f:Y \rightarrow T be continuous. Let \mathcal{M} be a countable base for the separable metrizable space T. By Lemma 2 Part 2, for each M \in \mathcal{M}, \overline{f^{-1}(M)} depends on countably many coordinates, say the countable set B_M. Let B=\bigcup_{M \in \mathcal{M}} B_M.

We claim that B is a countable set of coordinates we need. Let x,y \in Y such that x_\alpha=y_\alpha for all \alpha \in B. We need to show that f(x)=f(y). Suppose f(x) \ne f(y). Choose \left\{M_1,M_2,M_3,\cdots \right\} \subset \mathcal{M} such that

  • \left\{f(x) \right\}=\bigcap_{j=1}^\infty M_j=\bigcap_{j=1}^\infty \overline{M_j}
  • \overline{M_{j+1}} \subset M_j for each j

This is possible since T is a second countable space. Then f(y) \notin \overline{M_{k}} for some k. Furthermore, y \notin f^{-1}(\overline{M_{k}}). Since f is continuous, \overline{f^{-1}(M_{k})} \subset f^{-1}(\overline{M_{k}}). Therefore, y \notin \overline{f^{-1}(M_{k})}. On the other hand, \overline{f^{-1}(M_{k})} depends on the countably many coordinates in B_{M_k}. We assume above that x_\alpha=y_\alpha for all \alpha \in B. Thus x_\alpha=y_\alpha for all \alpha \in B_{M_k}. This means that y \in \overline{f^{-1}(M_{k})}, a contradiction. It must be that case that f(x)=f(y). \blacksquare

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Another Version

We state another version of Theorem 1 that will be useful in some situations.

Theorem 2

    Let X=\prod_{\alpha \in A} X_\alpha be a product space such that each factor X_\alpha is a separable space. Let T be a second countable space. Let Y be a dense subspace of X. Let f:Y \times Y \rightarrow T be any continuous function. Then the function f depends on countably many coordinates, which means either one of the following two conditions:

    1. There exists a countable set C \subset A such that for any (x,y),(p,q) \in Y \times Y, if x_\alpha=p_\alpha and y_\alpha=q_\alpha for all \alpha \in C, then f(x,y)=f(p,q).
    2. There exists a countable set C \subset A and there exists a continuous g:\pi_C(Y) \times \pi_C(Y) \rightarrow T such that f=g \circ (\pi_C \times \pi_C).

The map \pi_C \times \pi_C is the projection map from \prod_{\alpha \in A} X_\alpha \times \prod_{\alpha \in A} X_\alpha into the subproduct \prod_{\alpha \in C} X_\alpha \times \prod_{\alpha \in B} X_\alpha defined by (\pi_C \times \pi_C)(x,y)=(\pi_C(x),\pi_C(y)). In Theorem 2, we only need to consider \pi_C \times \pi_C being defined on the subspace Y \times Y.

Theorem 2 follows from Theorem 1. It is only a matter of fitting Theorem 2 in the framework of Theorem 1. Note that the product \prod_{\alpha \in A} X_\alpha \times \prod_{\alpha \in A} X_\alpha is identical to the product \prod_{\alpha \in A \cup A^*} X_\alpha where A^* is a disjoint copy of the index set A. For (x,y) \in \prod_{\alpha \in A} X_\alpha \times \prod_{\alpha \in A} X_\alpha, let x \times y \in \prod_{\alpha \in A \cup A^*} X_\alpha be defined by (x \times y)_\alpha=x_\alpha for all \alpha \in A and (x \times y)_\alpha=y_\alpha for all \alpha \in A^*.

With the identification of (x,y) with x \times y, we have a setting that fits Theorem 1. The product \prod_{\alpha \in A \cup A^*} X_\alpha is also a product of separable spaces. The set Y \times Y is a dense subspace of the product \prod_{\alpha \in A \cup A^*} X_\alpha. In this new setting, we view a point in Y \times Y as x \times y. The map f:Y \times Y \rightarrow T is still a continuous map. We can now apply Theorem 1.

Let B \subset A \cup A^* be a countable set such that for all x \times y,p \times q \in Y \times Y, if (x \times y)_\alpha=(p \times q)_\alpha for all \alpha \in B, then f(x \times y)=f(p \times q). Specifically, if x_\alpha=p_\alpha for all \alpha \in B \cap A and y_\alpha=q_\alpha for all \alpha \in B \cap A^*, then f(x,y)=f(p,q).

Choose a countable set C \subset A such that B \cap A \subset C and B \cap A^* \subset C^*. Here, C^* is the copy of C in A^*. We claim that C is a countable set we need in condition 1 of Theorem 2. Let (x,y),(p,q) \in Y \times Y such that x_\alpha=p_\alpha and y_\alpha=q_\alpha for all \alpha \in C. This implies that x_\alpha=p_\alpha for all \alpha \in B \cap A and y_\alpha=q_\alpha for all \alpha \in B \cap A^*. Then f(x,y)=f(p,q). Thus condition 1 of Theorem 2 holds. It is also straightforward to verify that Condition 1 and Condition 2 are equivalent.

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Remarks

The notion of factorizing a continuous map defined on a product space is an old topic. Theorem 1 discussed in this post is based on Theorem 4 found in [6]. Theorem 4 found in [6] is to factor continuous maps defined on a product of separable spaces. Theorem 1 in this post is modified to consider continuous maps defined on a dense subspace of a product of separable spaces. This modification will make it more useful. The references listed below represent a small sample of papers or books that have involves theorems of factoring functions defined on products. The work in [3] and [5] have more systematic treatment.

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Reference

  1. Brandenburg H., Husek M., On mappings from products into developable spaces, Topology Appl., 26, 229-238, 1987.
  2. Engelking R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.
  3. Engelking R., On functions defined on Cartesian products, Fund. Math., 59, 221-231, 1966.
  4. Keesling J., Normality and infinite product spaces, Adv. in. Math., 9, 90-92, 1972.
  5. Noble N., Ulmer M., Factoring functions on Cartesian products, Trans. Amer. Math. Soc., 163, 329-339, 1972.
  6. Ross K. A., Stone A. H., Products of separable spaces, Amer. Math. Monthly, 71, 398-403, 1964.

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\copyright \ 2014 \text{ by Dan Ma}

Another characterization about CCC spaces

In this post, we present another characterization about spaces with the countable chain condition (CCC spaces for short). The theorem presented here (Theorem 1 below) will provide more insight about CCC spaces and should be useful in proving theorems about CCC spaces. This characterization will make it easy to see that CCC spaces are weakly Lindelof.

This post can be considered a continuation of an earlier post, which discusses a different characterization of CCC spaces.

All spaces under consideration are at least T_1 and regular. A space X is said to have the countable chain condition (to have the CCC for short) if \mathcal{U} is a disjoint collection of non-empty open subsets of X (meaning that for any A,B \in \mathcal{U} with A \ne B, we have A \cap B=\varnothing), then \mathcal{U} is countable. In other words, in a space with the CCC, there cannot be uncountably many pairwise disjoint non-empty open sets. For ease of making a statement or stating a result, if X has the CCC, we also say that X is a CCC space or X is CCC. We prove the following theorem.

    Theorem 1

      Let X be a space. Then the following conditions are equivalent.

      1. The space has the CCC.
      2. For any collection \mathcal{U} of non-empty open subsets of X, there exists a countable \mathcal{V} \subset \mathcal{U} such that \bigcup \mathcal{U} \subset \overline{\bigcup \mathcal{V}}.

Proof of Theorem 1
1 \Longrightarrow 2
Suppose that condition 2 does not hold. Then there exists a collection \mathcal{U} of non-empty open subsets of X such that for any countable \mathcal{V} \subset \mathcal{U}, there exists a point x \in \bigcup \mathcal{U}=Y such that x \notin \overline{\bigcup \mathcal{V}}. From the collection \mathcal{U}, performing a transfinite inductive process, we will generate an uncountable collection of pairwise disjoint non-empty open subsets of X.

Choose some U_0 \in \mathcal{U}. Choose some x_0 \in Y-\overline{U_0}. For \alpha < \omega_1, suppose that the following have been chosen

    \left\{x_\beta \in Y: \beta<\alpha \right\}

    \left\{U_\beta \in \mathcal{U}: \beta<\alpha  \right\}

such that for each \beta<\alpha, x_\beta \notin \overline{\bigcup \limits_{\gamma < \beta} U_\gamma} and x_\beta \in U_\beta. Then by the assumption about \mathcal{U}, there exists x_\alpha \in Y such that x_\alpha \notin \overline{\bigcup \limits_{\gamma < \alpha} U_\gamma}. Now choose some U_\alpha \in \mathcal{U} such that x_\alpha \in U_\alpha. The inductive process is completed.

For each \alpha<\omega_1, let O_\alpha=U_\alpha-\overline{\bigcup \limits_{\gamma < \alpha} U_\gamma}. Clearly O_\alpha \ne \varnothing since x_\alpha \in O_\alpha. For \beta<\alpha<\omega_1, we have O_\beta \cap O_\alpha=\varnothing. With the non-empty open sets O_\alpha being pairwise disjoint, we conclude that X does not have the CCC.

2 \Longrightarrow 1
This is the easier direction. Suppose X is not CCC. Let \mathcal{W}=\left\{W_\alpha: \alpha<\omega_1 \right\} be a collection of pairwise disjoint non-empty open subsets of X. It is clear that for any countable \mathcal{V} \subset \mathcal{W}, the closure \overline{\bigcup \mathcal{V}} has to miss some W_\alpha (e.g. choose some W_\alpha \notin \mathcal{V}). Thus condition 2 does not hold. \blacksquare

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Weakly Lindelof Spaces

With Theorem 1, the CCC property looks like a covering property. Let X be a CCC space. Let \mathcal{U} be an open cover of X. By Theorem 1, there is a countable \mathcal{V} \subset \mathcal{U} such that X=\bigcup \mathcal{U} \subset \overline{\bigcup \mathcal{V}} (in other words, \bigcup \mathcal{V} is dense in X). So any CCC space X satisfies the following covering property:

    For any open cover \mathcal{U} of X, there exists a countable \mathcal{V} \subset \mathcal{U} such that \bigcup \mathcal{V} is dense in X.

Any space satisfying the above property is called a weakly Lindelof space. Any CCC space is weakly Lindelof. On the other hand, the weakly Lindelof property is strictly weaker than CCC. For a further discussion, see the next post called Weakly Lindelof spaces.

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\copyright \ 2014 \text{ by Dan Ma}

A theorem about CCC spaces

It is a well known result in general topology that in any regular space with the countable chain condition, paracompactness and the Lindelof property are equivalent. The proof of this result hinges on one theorem about the spaces with the countable chain condition. In this post we are to put the spotlight on this theorem (Theorem 1 below) and then use it to prove a few results. These results indicate that in a space with the countable chain condition with some weaker covering property is either Lindelof or paracompact.

This post is centered on a theorem about the CCC property (Theorem 1 and Theorem 1a below). So it can be considered as a continuation of a previous post on CCC called Some basic properties of spaces with countable chain condition. The results that are derived from Theorem 1 are also found in [2]. But the theorem concerning CCC is only a small part of that paper among several other focuses. In this post, the exposition is to explain several interesting theorems that are derived from Theorem 1. One of the theorems is the statement that every locally compact metacompact perfectly normal space is paracompact, a theorem originally proved by Arhangelskii (see Theorem 11 below).

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CCC Spaces

All spaces under consideration are at least T_1 and regular. A space X is said to have the countable chain condition (to have the CCC for short) if \mathcal{U} is a disjoint collection of non-empty open subsets of X (meaning that for any A,B \in \mathcal{U} with A \ne B, we have A \cap B=\varnothing), then \mathcal{U} is countable. In other words, in a space with the CCC, there cannot be uncountably many pairwise disjoint non-empty open sets. For ease of making a statement or stating a result, if X has the CCC, we also say that X is a CCC space or X is CCC.

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A Theorem about CCC Spaces

The theorem of CCC spaces we want to discuss has to do with collections of open sets that are “nice”. We first define what we mean by nice. Let \mathcal{A} be a collection of non-empty subsets of the space X. The collection \mathcal{A} is said to be point-finite (point-countable) if each point of X belongs to only finitely (countably) many sets in \mathcal{A}.

Now we define what we mean by “nice” collection of open sets. The collection \mathcal{A} is said to be locally finite (locally countable) at a point x \in X if there exists an open set O \subset X with x \in O such that O meets at most finitely (countably) many sets in \mathcal{A}. The collection \mathcal{A} is said to be locally finite (locally countable) if it is locally finite (locally countable) at each x \in X.

The property of being a separable space implies the CCC. The reverse is not true. However the CCC property is still a very strong property. The CCC property is equivalent to the property that if a collection of non-empty open sets is “nice” on a dense set of points, then the collection of open sets is a countable collection. The following is a precise statement.

    Theorem 1

      Let X be a CCC space. Then if \mathcal{U} is a collection of non-empty open subsets of X such that the following set

        D(\mathcal{U})=\left\{x \in X: \mathcal{U} \text{ is locally-countable at } x \right\}

      is dense in the open subspace \bigcup \mathcal{U}, then \mathcal{U} must be countable.

The collections of open sets in the above theorem do not have to be open covers. However, if they are open covers, the theorem can tie CCC spaces with some covering properties. As long as the space has the CCC, any open cover that is locally-countable on a dense set must be countable. Looking at it in the contrapositive angle, in a CCC space, any uncountable open cover is not locally-countable in some open set.

Proof of Theorem 1
Let \mathcal{U} be a collection of open subsets of X such that the set D(\mathcal{U}) as defined above is dense in the open subspace \bigcup \mathcal{U}. We show that \mathcal{U} is countable. Suppose not.

For each U \in \mathcal{U}, since U \cap D(\mathcal{U}) \ne \varnothing, we can choose a non-empty open set f(U) \subset U such that f(U) has non-empty intersection with only countably many sets in \mathcal{U}. Let \mathcal{U}_f be the following collection:

    \mathcal{U}_f=\left\{f(U): U \in \mathcal{U} \right\}

For H,K \in \mathcal{U}_f, by a chain from H to K, we mean a finite collection

    \left\{W_1,W_2,\cdots,W_n \right\} \subset \mathcal{U}_f

such that H=W_1, K=W_n and W_j \cap W_{j+1} \ne \varnothing for any 1 \le j <n. For each open set W \in \mathcal{U}_f, define \mathcal{C}(W) and \mathcal{E}(W) as follows:

    \mathcal{C}(W)=\left\{V \in \mathcal{U}_f: \text{there exists a chain from } W \text{ to } V \right\}

    \mathcal{E}(W)=\bigcup \mathcal{C}(W)

One observation we make is that for W_1,W_2 \in \mathcal{U}_f, if \mathcal{E}(W_1) \cap \mathcal{E}(W_2) \ne \varnothing, then \mathcal{C}(W_1)=\mathcal{C}(W_2) and \mathcal{E}(W_1)=\mathcal{E}(W_2). So the distinct \mathcal{E}(W) are pairwise disjoint. Because the space X has the CCC, there can be only countably many distinct open sets \mathcal{E}(W). Thus there can be only countably many distinct collections \mathcal{C}(W).

Note that each \mathcal{C}(W) is a countable collection of open sets. Each V \in \mathcal{U}_f meets only countably many open sets in \mathcal{U}. So each V \in \mathcal{U}_f can meet only countably many sets in \mathcal{U}_f, since for each V \in \mathcal{U}_f, V \subset U for some U \in \mathcal{U}. Thus for each W \in \mathcal{U}_f, in considering all finite-length chain starting from W, there can be only countably many open sets in \mathcal{U}_f that can be linked to W. Thus \mathcal{C}(W) must be countable. In taking the union of all \mathcal{C}(W), we get back the collection \mathcal{U}_f. Thus we have:

    \mathcal{U}_f=\bigcup \limits_{W \in \mathcal{U}_f} \mathcal{C}(W)

Because the space X is CCC, there are only countably many distinct collections \mathcal{C}(W) in the above union. Each \mathcal{C}(W) is countable. So \mathcal{U}_f is a countable collection of open sets.

Furthermore, each U \in \mathcal{U} contains at least one set in \mathcal{U}_f. From the way we choose sets in \mathcal{U}_f, we see that for each V \in \mathcal{U}_f, V=f(U) \subset U for at most countably many U \in \mathcal{U}. The argument indicates that we have a one-to-countable mapping from \mathcal{U}_f to \mathcal{U}. Thus the original collection \mathcal{U} must be countable. \blacksquare

The property in Theorem 1 is actually equivalent to the CCC property. Just that the proof of Theorem 1 represents the hard direction that needs to be proved. Theorem 1 can be expanded to be the following theorem.

    Theorem 1a

      Let X be a space. Then the following conditions are equivalent.

      1. The space X has the CCC.
      2. If \mathcal{U} is a collection of non-empty open subsets of X such that the following set

          D(\mathcal{U})=\left\{x \in X: \mathcal{U} \text{ is locally-countable at } x \right\}

        is dense in the open subspace \bigcup \mathcal{U}, then \mathcal{U} must be countable.

      3. If \mathcal{U} is a collection of non-empty open subsets of X such that \mathcal{U} is locally-countable at every point in the open subspace \bigcup \mathcal{U}, then \mathcal{U} must be countable.

The direction 1 \rightarrow 2 has been proved above. The directions 2 \rightarrow 3 and 3 \rightarrow 1 are straightforward.

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Tying Theorem 1 to “Nice” Open Covers

One easy application of Theorem 1 is to tie it to locally-finite and locally-countable open covers. We have the following theorem.

    Theorem 2

      In any CCC space, any locally-countable open cover must be countable. Thus any locally-finite open cover must also be countable.

Theorem 2 gives the well known result that any CCC paracompact space is Lindelof (see Theorem 5 below). In fact, Theorem 2 gives the result that any CCC para-Lindelof space is Lindelof (see Theorem 6 below). A space X is para-Lindelof if every open cover has a locally-countable open refinement.

Can Theorem 2 hold for point-finite covers (or point-countable covers)? The answer is no (see Example 1 below). With the additional property of having a Baire space, we have the following theorem.

    Theorem 3

      In any Baire space with the CCC, any point-finite open cover must be countable.

A Space X is a Baire space if U_1,U_2,U_3,\cdots are dense open subsets of X, then \bigcap \limits_{j=1}^\infty U_j \ne \varnothing. For more information about Baire spaces, see this previous post.
.

Proof of Theorem 3
Let X be a Baire space with the CCC. Let \mathcal{U} be a point-finite open cover of X. Suppose that \mathcal{U} is uncountable. We show that this assumption with lead to a contradiction. Thus \mathcal{U} must be countable.

By Theorem 1, there exists an open set V \subset X such that \mathcal{U} is not locally-countable at any point in V. For each positive integer n, let H_n be the following:

    H_n=\left\{x \in V: x \text{ is in at most } n \text{ sets in } \mathcal{U} \right\}

Note that V=\bigcup \limits_{j=1}^\infty H_j. Furthermore, each H_n is a closed set in the space V. Since X is a Baire space, every non-empty open subset of X is of second category (i.e. it cannot be a union of countably many closed and nowhere dense sets). Thus it cannot be that each H_n is nowhere dense in V. For some n, H_n is not nowhere dense. There must exist some open W \subset V such that H_n \cap W is dense in W. Because H_n is closed, W \subset H_n.

Choose y \in W. The point y is in at most n open sets in \mathcal{U}. Let U_1,U_2,\cdots,U_m \in \mathcal{U} such that y \in \bigcap \limits_{j=1}^m U_j. Clearly 1 \le m \le n. Let U=W \cap U_1 \cap \cdots \cap U_m. Note that y \in U \subset H_n \subset V.

Every point in U belongs to at most n many sets in \mathcal{U} and already belong to m sets in \mathcal{U}. So each point in U can belong to at most n-m additional open sets in \mathcal{U}. Consider the case n-m=0 and the case n-m>0. We show that each case leads to a contradiction.

Suppose that n-m=0. Then each point of U can only meet n open sets in \mathcal{U}, namely U_1,U_2,\cdots,U_m. This contradicts that \mathcal{U} is not locally-countable at points in U \subset V.

Suppose that k=n-m>0. Let \mathcal{U}^*=\mathcal{U}-\left\{U_1,\cdots,U_m \right\}. Let \mathcal{M} be the following collection:

    \mathcal{M}=\left\{U \cap \bigcap \limits_{O \in M} O \ne \varnothing: M \subset \mathcal{U}^* \text{ and } \lvert M \lvert=k \right\}

Each element of \mathcal{M} is an open subset of U that is the intersection of exactly n many open sets in \mathcal{U}. So \mathcal{M} is a collection of pairwise disjoint open sets. The open set U as a topological space has the CCC. So \mathcal{M} is at most countable. Thus the open set U meets at most countably many open sets in \mathcal{U}, contradicting that \mathcal{U} is not locally-countable at points in U \subset V.

Both cases n-m=0 and n-m>0 lead to contradiction. So \mathcal{U} must be countable. The proof to Theorem 3 is completed. \blacksquare

As a corollary to Theorem 3, we have the result that every Baire CCC metacompact space is Lindelof.

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Some Applications of Theorems 2 and 3

In proving paracompactness in some of the theorems, we need a theorem involving the concept of star-countable open cover. A collection \mathcal{A} of subsets of a space X is said to be star-finite (star-countable) if for each A \in \mathcal{A}, only finitely (countably) many sets in \mathcal{A} meets A, i.e., the following set

    \left\{B \in \mathcal{A}: B \cap A \ne \varnothing \right\}

is finite (countable). The proof of the following theorem can be found in Engleking (see the direction (iv) implies (i) in the proof of Theorem 5.3.10 on page 326 in [1]).

    Theorem 4

      If every open cover of a regular space X has a star-countable open refinement, then X is paracompact.

As indicated in the above section, Theorem 2 and Theorem 3 have some obvious applications. We have the following theorems.

    Theorem 5

      Let X be a CCC space. Then X is paracompact if and only of X is Lindelof.

Proof of Theorem 5
The direction \Longleftarrow follows from the fact that any regular Lindelof space is paracompact.

The direction \Longrightarrow follows from Theorem 2. \blacksquare

    Theorem 6

      Every CCC para-Lindelof space is Lindelof.

Proof of Theorem 6
This also follows from Theorem 2. \blacksquare

    Theorem 7

      Every Baire CCC metacompact space is Lindelof.

Proof of Theorem 7
Let X be a Baire CCC metacompact space. Let \mathcal{U} be an open cover of X. By metacompactness, let \mathcal{V} be a point-finite open refinement of \mathcal{U}. By Theorem 3, \mathcal{V} must be countable. \blacksquare

    Theorem 8

      Every Baire CCC hereditarily metacompact space is hereditarily Lindelof.

Proof of Theorem 8
Let X be a Baire CCC hereditarily metacompact space. To show that X is hereditarily Lindelof, it suffices to show that every non-empty open subset is Lindelof. Let Y \subset X be open. Then Y has the CCC and is also metacompact. Being a Baire space is hereditary with respect to open subspaces. So Y is a Baire space too. By Theorem 7, Y is Lindelof. \blacksquare

    Theorem 9

      Every locally CCC regular para-Lindelof space is paracompact.

Proof of Theorem 9
A space is locally CCC if every point has an open neighborhood that has the CCC. Let X be a regular space that is locally CCC and para-Lindelof. Let \mathcal{U} be an open cover of X. Using the locally CCC assumption and by taking a refinement of \mathcal{U} if necessary, we can assume that each open set in \mathcal{U} has the CCC. By the para-Lindelof assumption, let \mathcal{V} be a locally-countable open refinement of \mathcal{U}. So each open set in \mathcal{V} has the CCC too.

Now we show that \mathcal{V} is star-countable. Let V \in \mathcal{V}. Let \mathcal{G} be the following collection:

    \mathcal{G}=\left\{V \cap W: W \in \mathcal{V} \right\}

which is is open cover of V. Within the subspace V, \mathcal{G} is a locally-countable open cover. By Theorem 2, \mathcal{G} must be countable. The collection \mathcal{G} represents all the open sets in \mathcal{V} that have non-empty intersection with V. Thus only countably many open sets in \mathcal{V} can meet V. So \mathcal{V} is a star-countable open refinement of \mathcal{U}. By Theorem 4, X is paracompact. \blacksquare

    Theorem 10

      Every locally CCC regular metacompact Baire space is paracompact.

Proof of Theorem 10
Let X be a regular space that is locally CCC and is a metacompact Baire space. Let \mathcal{U} be an open cover of X. Using the locally CCC assumption and by taking a refinement of \mathcal{U} if necessary, we can assume that each open set in \mathcal{U} has the CCC. By the metacompact assumption, let \mathcal{V} be a point-finite open refinement of \mathcal{U}. So each open set in \mathcal{V} has the CCC too. Each open set in \mathcal{V} is also a Baire space.

Now we show that \mathcal{V} is star-countable. Let V \in \mathcal{V}. Let \mathcal{G} be the following collection:

    \mathcal{G}=\left\{V \cap W: W \in \mathcal{V} \right\}

which is is open cover of V. Within the subspace V, \mathcal{G} is a point-finite open cover. By Theorem 3, \mathcal{G} must be countable. The collection \mathcal{G} represents all the open sets in \mathcal{V} that have non-empty intersection with V. Thus only countably many open sets in \mathcal{V} can meet V. So \mathcal{V} is a star-countable open refinement of \mathcal{U}. By Theorem 4, X is paracompact. \blacksquare

    Theorem 11

      Every locally compact metacompact perfectly normal space is paracompact.

Proof of Theorem 11
This follows from Theorem 10 after we prove the following two points:

  • Any locally compact space is a Baire space.
  • Any perfect locally compact space is locally CCC.

To see the first point, let Y be a locally compact space. Let W_1,W_2,W_3,\cdots be dense open sets in Y. Let y \in Y and let W \subset Y be open such that y \in W and \overline{W} is compact. We show that W contains a point that belongs to all W_n. Let X_1=W \cap W_1, which is open and non-empty. Next choose non-empty open X_2 such that \overline{X_2} \subset X_1 and X_2 \subset W_2. Next choose non-empty open X_3 such that \overline{X_3} \subset X_2 and X_3 \subset W_3. Continue in this manner, we have a sequence of open sets X_1,X_2,X_3,\cdots such that for each n, \overline{X_{n+1}} \subset X_n and \overline{X_n} is compact. The intersection of all the X_n is non-empty. The points in the intersection must belong to each W_n.

To see the second point, let Y be a locally compact space such that every closed set is a G_\delta-set. Suppose that Y is not locally CCC at y \in Y. Let U \subset Y be open such that y \in U and \overline{U} is compact. Then U must not have the CCC. Let \left\{U_\alpha: \alpha<\omega_1 \right\} be a pairwise disjoint collection of open subsets of U. Let O=\bigcup \limits_{\alpha<\omega_1} U_\alpha and let C=Y-O.

Let C=\bigcap \limits_{n=1}^\infty V_n where each V_n is open in Y and V_{n+1} \subset V_n for each integer n. For each \alpha<\omega_1, pick y_\alpha \in U_\alpha. For each y_\alpha, there is some integer f(\alpha) such that y_\alpha \notin V_{f(\alpha)}. So there must exist some integer n such that A=\left\{y_\alpha: f(\alpha)=n \right\} is uncountable.

The set A is an infinite subset of the compact set \overline{U}. So A has a limit point, say p (also called cluster point). Clearly p \notin O. So p \in C. In particular, p \in V_n. Then V_n contains some points of A. But for any y_\alpha \in A, y_\alpha \notin V_n=V_{f(\alpha)}, a contradiction. So Y must be locally CCC at each y \in Y. \blacksquare

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Some Examples

Example 1
A CCC space X with an uncountable point-finite open covers. This example demonstrates that in Theorem 2, locally-finite or locally-countable cannot be replaced by point-finite. Consider the following product space:

    Y=\prod \limits_{\alpha < \omega_1} \left\{0,1 \right\}=\left\{0,1 \right\}^{\omega_1}

i.e, the product space of \omega_1 many copies of the two-point discrete space \left\{0,1 \right\}. Let X be the set of all points h \in Y such that h(\alpha)=1 for only finitely many \alpha<\omega_1.

The product space Y is the product of separable spaces, hence has the CCC. The space X is dense in Y. Hence X has the CCC. For each \alpha<\omega_1, define U_\alpha as follows:

    U_\alpha=\left\{h \in X: h(\alpha)=1 \right\}

Then \left\{U_\alpha:\alpha<\omega_1  \right\} is a point-finite open cover of X. Of course, X in this example is not a Baire space. \blacksquare

The following three examples center around the four properties in Theorem 7 (Baire + CCC + metacompact imply Lindelof). These examples show that each property in the hypothesis is crucial.

Example 2
A separable non-Lindelof space that is a Baire space. This example shows that the metacompact assumption is crucial for Theorem 7.

The example is the Sorgenfrey plane S \times S where S is the real line with the Sorgenfrey topology (generated by the half-open intervals of the form [a,b)). It is well known that S \times S is not Lindelof. The Sorgenfrey plane is Baire and is separable (hence CCC). Furthermore, S \times S is not metacompact (if it were, it would be Lindelof by Theorem 7). \blacksquare

Example 3
A non-Lindelof metacompact Baire space M. This example shows that the CCC assumption in Theorem 7 is necessary.

This space M is the subspace of Bing’s Example G that has finite support (defined and discussed in the post A subspace of Bing’s example G. It is normal and not collectionwise normal (hence cannot be Lindelof) and metacompact. The space M does not have CCC since it has uncountably many isolated points. Any space with a dense set of isolated points is a Baire space. Thus the space M is also a Baire space. \blacksquare

Example 4
A non-Lindelof CCC metacompact non-Baire space W. This example shows that the Baire space assumption in Theorem 7 is necessary.

Let W be the set of all non-empty finite subsets of the real line with the Pixley-Roy topology. Note that W is non-Lindelof and has the CCC and is metacompact. Of course it is not Baire. For more information on Pixley-Roy spaces, see the post called Pixley-Roy hyperspaces. For the purpose of this example, the Pixley-Roy space can be built on any uncountable separable metrizable space.

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Reference

  1. Engelking, R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.
  2. Tall, F. D., The Countable Chain Condition Versus Separability – Applications of Martin’s Axiom, Gen. Top. Appl., 4, 315-339, 1974.

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\copyright \ 2014 \text{ by Dan Ma}

Pixley-Roy hyperspaces

In this post, we introduce a class of hyperspaces called Pixley-Roy spaces. This is a well-known and well studied set of topological spaces. Our goal here is not to be comprehensive but rather to present some selected basic results to give a sense of what Pixley-Roy spaces are like.

A hyperspace refers to a space in which the points are subsets of a given “ground” space. There are more than one way to define a hyperspace. Pixley-Roy spaces were first described by Carl Pixley and Prabir Roy in 1969 (see [5]). In such a space, the points are the non-empty finite subsets of a given ground space. More precisely, let X be a T_1 space (i.e. finite sets are closed). Let \mathcal{F}[X] be the set of all non-empty finite subsets of X. For each F \in \mathcal{F}[X] and for each open subset U of X with F \subset U, we define:

    [F,U]=\left\{B \in \mathcal{F}[X]: F \subset B \subset U \right\}

The sets [F,U] over all possible F and U form a base for a topology on \mathcal{F}[X]. This topology is called the Pixley-Roy topology (or Pixley-Roy hyperspace topology). The set \mathcal{F}[X] with this topology is called a Pixley-Roy space.

The hyperspace as defined above was first defined by Pixley and Roy on the real line (see [5]) and was later generalized by van Douwen (see [7]). These spaces are easy to define and is useful for constructing various kinds of counterexamples. Pixley-Roy played an important part in answering the normal Moore space conjecture. Pixley-Roy spaces have also been studied in their own right. Over the years, many authors have investigated when the Pixley-Roy spaces are metrizable, normal, collectionwise Hausdorff, CCC and homogeneous. For a small sample of such investigations, see the references listed at the end of the post. Our goal here is not to discuss the results in these references. Instead, we discuss some basic properties of Pixley-Roy to solidify the definition as well as to give a sense of what these spaces are like. Good survey articles of Pixley-Roy are [3] and [7].

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Basic Discussion

In this section, we focus on properties that are always possessed by a Pixley-Roy space given that the ground space is at least T_1. Let X be a T_1 space. We discuss the following points:

  1. The topology defined above is a legitimate one, i.e., the sets [F,U] indeed form a base for a topology on \mathcal{F}[X].
  2. \mathcal{F}[X] is a Hausdorff space.
  3. \mathcal{F}[X] is a zero-dimensional space.
  4. \mathcal{F}[X] is a completely regular space.
  5. \mathcal{F}[X] is a hereditarily metacompact space.

Let \mathcal{B}=\left\{[F,U]: F \in \mathcal{F}[X] \text{ and } U \text{ is open in } X \right\}. Note that every finite set F belongs to at least one set in \mathcal{B}, namely [F,X]. So \mathcal{B} is a cover of \mathcal{F}[X]. For A \in [F_1,U_1] \cap [F_2,U_2], we have A \in [A,U_1 \cap U_2] \subset   [F_1,U_1] \cap [F_2,U_2]. So \mathcal{B} is indeed a base for a topology on \mathcal{F}[X].

To show \mathcal{F}[X] is Hausdorff, let A and B be finite subsets of X where A \ne B. Then one of the two sets has a point that is not in the other one. Assume we have x \in A-B. Since X is T_1, we can find open sets U, V \subset X such that x \in U, x \notin V and A \cup B-\left\{ x \right\} \subset V. Then [A,U \cup V] and [B,V] are disjoint open sets containing A and B respectively.

To see that \mathcal{F}[X] is a zero-dimensional space, we show that \mathcal{B} is a base consisting of closed and open sets. To see that [F,U] is closed, let C \notin [F,U]. Either F \not \subset C or C \not \subset U. In either case, we can choose open V \subset X with C \subset V such that [C,V] \cap [F,U]=\varnothing.

The fact that \mathcal{F}[X] is completely regular follows from the fact that it is zero-dimensional.

To show that \mathcal{F}[X] is metacompact, let \mathcal{G} be an open cover of \mathcal{F}[X]. For each F \in \mathcal{F}[X], choose G_F \in \mathcal{G} such that F \in G_F and let V_F=[F,X] \cap G_F. Then \mathcal{V}=\left\{V_F: F \in \mathcal{F}[X] \right\} is a point-finite open refinement of \mathcal{G}. For each A \in \mathcal{F}[X], A can only possibly belong to V_F for the finitely many F \subset A.

A similar argument show that \mathcal{F}[X] is hereditarily metacompact. Let Y \subset \mathcal{F}[X]. Let \mathcal{H} be an open cover of Y. For each F \in Y, choose H_F \in \mathcal{H} such that F \in H_F and let W_F=([F,X] \cap Y) \cap H_F. Then \mathcal{W}=\left\{W_F: F \in Y \right\} is a point-finite open refinement of \mathcal{H}. For each A \in Y, A can only possibly belong to W_F for the finitely many F \subset A such that F \in Y.

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More Basic Results

We now discuss various basic topological properties of \mathcal{F}[X]. We first note that \mathcal{F}[X] is a discrete space if and only if the ground space X is discrete. Though we do not need to make this explicit, it makes sense to focus on non-discrete spaces X when we look at topological properties of \mathcal{F}[X]. We discuss the following points:

  1. If X is uncountable, then \mathcal{F}[X] is not separable.
  2. If X is uncountable, then every uncountable subspace of \mathcal{F}[X] is not separable.
  3. If \mathcal{F}[X] is Lindelof, then X is countable.
  4. If \mathcal{F}[X] is Baire space, then X is discrete.
  5. If \mathcal{F}[X] has the CCC, then X has the CCC.
  6. If \mathcal{F}[X] has the CCC, then X has no uncountable discrete subspaces,i.e., X has countable spread, which of course implies CCC.
  7. If \mathcal{F}[X] has the CCC, then X is hereditarily Lindelof.
  8. If \mathcal{F}[X] has the CCC, then X is hereditarily separable.
  9. If X has a countable network, then \mathcal{F}[X] has the CCC.
  10. The Pixley-Roy space of the Sorgenfrey line does not have the CCC.
  11. If X is a first countable space, then \mathcal{F}[X] is a Moore space.

Bullet points 6 to 9 refer to properties that are never possessed by Pixley-Roy spaces except in trivial cases. Bullet points 6 to 8 indicate that \mathcal{F}[X] can never be separable and Lindelof as long as the ground space X is uncountable. Note that \mathcal{F}[X] is discrete if and only if X is discrete. Bullet point 9 indicates that any non-discrete \mathcal{F}[X] can never be a Baire space. Bullet points 10 to 13 give some necessary conditions for \mathcal{F}[X] to be CCC. Bullet 14 gives a sufficient condition for \mathcal{F}[X] to have the CCC. Bullet 15 indicates that the hereditary separability and the hereditary Lindelof property are not sufficient conditions for the CCC of Pixley-Roy space (though they are necessary conditions). Bullet 16 indicates that the first countability of the ground space is a strong condition, making \mathcal{F}[X] a Moore space.

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To see bullet point 6, let X be an uncountable space. Let \left\{F_1,F_2,F_3,\cdots \right\} be any countable subset of \mathcal{F}[X]. Choose a point x \in X that is not in any F_n. Then none of the sets F_i belongs to the basic open set [\left\{x \right\} ,X]. Thus \mathcal{F}[X] can never be separable if X is uncountable.

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To see bullet point 7, let Y \subset \mathcal{F}[X] be uncountable. Let W=\cup \left\{F: F \in Y \right\}. Let \left\{F_1,F_2,F_3,\cdots \right\} be any countable subset of Y. We can choose a point x \in W that is not in any F_n. Choose some A \in Y such that x \in A. Then none of the sets F_n belongs to the open set [A ,X] \cap Y. So not only \mathcal{F}[X] is not separable, no uncountable subset of \mathcal{F}[X] is separable if X is uncountable.

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To see bullet point 8, note that \mathcal{F}[X] has no countable open cover consisting of basic open sets, assuming that X is uncountable. Consider the open collection \left\{[F_1,U_1],[F_2,U_2],[F_3,U_3],\cdots \right\}. Choose x \in X that is not in any of the sets F_n. Then \left\{ x \right\} cannot belong to [F_n,U_n] for any n. Thus \mathcal{F}[X] can never be Lindelof if X is uncountable.

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For an elementary discussion on Baire spaces, see this previous post.

To see bullet point 9, let X be a non-discrete space. To show \mathcal{F}[X] is not Baire, we produce an open subset that is of first category (i.e. the union of countably many closed nowhere dense sets). Let x \in X a limit point (i.e. an non-isolated point). We claim that the basic open set V=[\left\{ x \right\},X] is a desired open set. Note that V=\bigcup \limits_{n=1}^\infty H_n where

    H_n=\left\{F \in \mathcal{F}[X]: x \in F \text{ and } \lvert F \lvert \le n \right\}

We show that each H_n is closed and nowhere dense in the open subspace V. To see that it is closed, let A \notin H_n with x \in A. We have \lvert A \lvert>n. Then [A,X] is open and every point of [A,X] has more than n points of the space X. To see that H_n is nowhere dense in V, let [B,U] be open with [B,U] \subset V. It is clear that x \in B \subset U where U is open in the ground space X. Since the point x is not an isolated point in the space X, U contains infinitely many points of X. So choose an finite set C with at least 2 \times n points such that B \subset C \subset U. For the the open set [C,U], we have [C,U] \subset [B,U] and [C,U] contains no point of H_n. With the open set V being a union of countably many closed and nowhere dense sets in V, the open set V is not of second category. We complete the proof that \mathcal{F}[X] is not a Baire space.

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To see bullet point 10, let \mathcal{O} be an uncountable and pairwise disjoint collection of open subsets of X. For each O \in \mathcal{O}, choose a point x_O \in O. Then \left\{[\left\{ x_O \right\},O]: O \in \mathcal{O} \right\} is an uncountable and pairwise disjoint collection of open subsets of \mathcal{F}[X]. Thus if \mathcal{F}[X] is CCC then X must have the CCC.

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To see bullet point 11, let Y \subset X be uncountable such that Y as a space is discrete. This means that for each y \in Y, there exists an open O_y \subset X such that y \in O_y and O_y contains no point of Y other than y. Then \left\{[\left\{y \right\},O_y]: y \in Y \right\} is an uncountable and pairwise disjoint collection of open subsets of \mathcal{F}[X]. Thus if \mathcal{F}[X] has the CCC, then the ground space X has no uncountable discrete subspace (such a space is said to have countable spread).

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To see bullet point 12, let Y \subset X be uncountable such that Y is not Lindelof. Then there exists an open cover \mathcal{U} of Y such that no countable subcollection of \mathcal{U} can cover Y. We can assume that sets in \mathcal{U} are open subsets of X. Also by considering a subcollection of \mathcal{U} if necessary, we can assume that cardinality of \mathcal{U} is \aleph_1 or \omega_1. Now by doing a transfinite induction we can choose the following sequence of points and the following sequence of open sets:

    \left\{x_\alpha \in Y: \alpha < \omega_1 \right\}

    \left\{U_\alpha \in \mathcal{U}: \alpha < \omega_1 \right\}

such that x_\beta \ne x_\gamma if \beta \ne \gamma, x_\alpha \in U_\alpha and x_\alpha \notin \bigcup \limits_{\beta < \alpha} U_\beta for each \alpha < \omega_1. At each step \alpha, all the previously chosen open sets cannot cover Y. So we can always choose another point x_\alpha of Y and then choose an open set in \mathcal{U} that contains x_\alpha.

Then \left\{[\left\{x_\alpha \right\},U_\alpha]: \alpha < \omega_1 \right\} is a pairwise disjoint collection of open subsets of \mathcal{F}[X]. Thus if \mathcal{F}[X] has the CCC, then X must be hereditarily Lindelof.

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To see bullet point 13, let Y \subset X. Consider open sets [A,U] where A ranges over all finite subsets of Y and U ranges over all open subsets of X with A \subset U. Let \mathcal{G} be a collection of such [A,U] such that \mathcal{G} is pairwise disjoint and \mathcal{G} is maximal (i.e. by adding one more open set, the collection will no longer be pairwise disjoint). We can apply a Zorn lemma argument to obtain such a maximal collection. Let D be the following subset of Y.

    D=\bigcup \left\{A: [A,U] \in \mathcal{G} \text{ for some open } U  \right\}

We claim that the set D is dense in Y. Suppose that there is some open set W \subset X such that W \cap Y \ne \varnothing and W \cap D=\varnothing. Let y \in W \cap Y. Then [\left\{y \right\},W] \cap [A,U]=\varnothing for all [A,U] \in \mathcal{G}. So adding [\left\{y \right\},W] to \mathcal{G}, we still get a pairwise disjoint collection of open sets, contradicting that \mathcal{G} is maximal. So D is dense in Y.

If \mathcal{F}[X] has the CCC, then \mathcal{G} is countable and D is a countable dense subset of Y. Thus if \mathcal{F}[X] has the CCC, the ground space X is hereditarily separable.

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A collection \mathcal{N} of subsets of a space Y is said to be a network for the space Y if any non-empty open subset of Y is the union of elements of \mathcal{N}, equivalently, for each y \in Y and for each open U \subset Y with y \in U, there is some A \in \mathcal{N} with x \in A \subset U. Note that a network works like a base but the elements of a network do not have to be open. The concept of network and spaces with countable network are discussed in these previous posts Network Weight of Topological Spaces – I and Network Weight of Topological Spaces – II.

To see bullet point 14, let \mathcal{N} be a network for the ground space X such that \mathcal{N} is also countable. Assume that \mathcal{N} is closed under finite unions (for example, adding all the finite unions if necessary). Let \left\{[A_\alpha,U_\alpha]: \alpha < \omega_1 \right\} be a collection of basic open sets in \mathcal{F}[X]. Then for each \alpha, find B_\alpha \in \mathcal{N} such that A_\alpha \subset B_\alpha \subset U_\alpha. Since \mathcal{N} is countable, there is some B \in \mathcal{N} such that M=\left\{\alpha< \omega_1: B=B_\alpha \right\} is uncountable. It follows that for any finite E \subset M, \bigcap \limits_{\alpha \in E} [A_\alpha,U_\alpha] \ne \varnothing.

Thus if the ground space X has a countable network, then \mathcal{F}[X] has the CCC.

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The implications in bullet points 12 and 13 cannot be reversed. Hereditarily Lindelof property and hereditarily separability are not sufficient conditions for \mathcal{F}[X] to have the CCC. See [4] for a study of the CCC property of the Pixley-Roy spaces.

To see bullet point 15, let S be the Sorgenfrey line, i.e. the real line \mathbb{R} with the topology generated by the half closed intervals of the form [a,b). For each x \in S, let U_x=[x,x+1). Then \left\{[ \left\{ x \right\},U_x]: x \in S \right\} is a collection of pairwise disjoint open sets in \mathcal{F}[S].

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A Moore space is a space with a development. For the definition, see this previous post.

To see bullet point 16, for each x \in X, let \left\{B_n(x): n=1,2,3,\cdots \right\} be a decreasing local base at x. We define a development for the space \mathcal{F}[X].

For each finite F \subset X and for each n, let B_n(F)=\bigcup \limits_{x \in F} B_n(x). Clearly, the sets B_n(F) form a decreasing local base at the finite set F. For each n, let \mathcal{H}_n be the following collection:

    \mathcal{H}_n=\left\{[F,B_n(F)]: F \in \mathcal{F}[X] \right\}

We claim that \left\{\mathcal{H}_n: n=1,2,3,\cdots \right\} is a development for \mathcal{F}[X]. To this end, let V be open in \mathcal{F}[X] with F \in V. If we make n large enough, we have [F,B_n(F)] \subset V.

For each non-empty proper G \subset F, choose an integer f(G) such that [F,B_{f(G)}(F)] \subset V and F \not \subset B_{f(G)}(G). Let m be defined by:

    m=\text{max} \left\{f(G): G \ne \varnothing \text{ and } G \subset F \text{ and } G \text{ is proper} \right\}

We have F \not \subset B_{m}(G) for all non-empty proper G \subset F. Thus F \notin [G,B_m(G)] for all non-empty proper G \subset F. But in \mathcal{H}_m, the only sets that contain F are [F,B_m(F)] and [G,B_m(G)] for all non-empty proper G \subset F. So [F,B_m(F)] is the only set in \mathcal{H}_m that contains F, and clearly [F,B_m(F)] \subset V.

We have shown that for each open V in \mathcal{F}[X] with F \in V, there exists an m such that any open set in \mathcal{H}_m that contains F must be a subset of V. This shows that the \mathcal{H}_n defined above form a development for \mathcal{F}[X].

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Examples

In the original construction of Pixley and Roy, the example was \mathcal{F}[\mathbb{R}]. Based on the above discussion, \mathcal{F}[\mathbb{R}] is a non-separable CCC Moore space. Because the density (greater than \omega for not separable) and the cellularity (=\omega for CCC) do not agree, \mathcal{F}[\mathbb{R}] is not metrizable. In fact, it does not even have a dense metrizable subspace. Note that countable subspaces of \mathcal{F}[\mathbb{R}] are metrizable but are not dense. Any uncountable dense subspace of \mathcal{F}[\mathbb{R}] is not separable but has the CCC. Not only \mathcal{F}[\mathbb{R}] is not metrizable, it is not normal. The problem of finding X \subset \mathbb{R} for which \mathcal{F}[X] is normal requires extra set-theoretic axioms beyond ZFC (see [6]). In fact, Pixley-Roy spaces played a large role in the normal Moore space conjecture. Assuming some extra set theory beyond ZFC, there is a subset M \subset \mathbb{R} such that \mathcal{F}[M] is a CCC metacompact normal Moore space that is not metrizable (see Example I in [8]).

On the other hand, Pixley-Roy space of the Sorgenfrey line and the Pixley-Roy space of \omega_1 (the first uncountable ordinal with the order topology) are metrizable (see [3]).

The Sorgenfrey line and the first uncountable ordinal are classic examples of topological spaces that demonstrate that topological spaces in general are not as well behaved like metrizable spaces. Yet their Pixley-Roy spaces are nice. The real line and other separable metric spaces are nice spaces that behave well. Yet their Pixley-Roy spaces are very much unlike the ground spaces. This inverse relation between the ground space and the Pixley-Roy space was noted by van Douwen (see [3] and [7]) and is one reason that Pixley-Roy hyperspaces are a good source of counterexamples.

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Reference

  1. Bennett, H. R., Fleissner, W. G., Lutzer, D. J., Metrizability of certain Pixley-Roy spaces, Fund. Math. 110, 51-61, 1980.
  2. Daniels, P, Pixley-Roy Spaces Over Subsets of the Reals, Topology Appl. 29, 93-106, 1988.
  3. Lutzer, D. J., Pixley-Roy topology, Topology Proc. 3, 139-158, 1978.
  4. Hajnal, A., Juahasz, I., When is a Pixley-Roy Hyperspace CCC?, Topology Appl. 13, 33-41, 1982.
  5. Pixley, C., Roy, P., Uncompletable Moore spaces, Proc. Auburn Univ. Conf. Auburn, AL, 1969.
  6. Przymusinski, T., Normality and paracompactness of Pixley-Roy hyperspaces, Fund. Math. 113, 291-297, 1981.
  7. van Douwen, E. K., The Pixley-Roy topology on spaces of subsets, Set-theoretic Topology, Academic Press, New York, 111-134, 1977.
  8. Tall, F. D., Normality versus Collectionwise Normality, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds), Elsevier Science Publishers B. V., Amsterdam, 685-732, 1984.
  9. Tanaka, H, Normality and hereditary countable paracompactness of Pixley-Roy hyperspaces, Fund. Math. 126, 201-208, 1986.

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\copyright \ 2014 \text{ by Dan Ma}