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}

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Product Space – Exercise Set 1

This post presents several exercises concerning product spaces. All the concepts involved in the exercises have been discussed in the blog.

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Exercise 1

Exercise 1a
Prove or disprove:
If X and Y are both hereditarily separable, then X \times Y is hereditarily separable.

Exercise 1b
Show that if each X_\alpha is separable, then the product space \prod_{\alpha < \omega} \ X_\alpha is separable.

Exercise 1c
Prove or disprove:
If each X_\alpha is separable, then the product space \prod_{\alpha < \omega_1} \ X_\alpha is not separable.

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Exercise 2

Exercise 2a
Show that if the space X is normal, then every closed subspace of X is a normal space.

Exercise 2b
Prove or disprove:
If the space X is normal, then every dense open subspace of X is a normal space.

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Exercise 3

Consider the product space \prod_{\alpha \in W} \ X_\alpha.

Exercise 3a
Suppose that X_\alpha is compact for all but one \alpha \in W such that the non-compact factor is a Lindelof space. Show that the product space \prod_{\alpha \in W} \ X_\alpha is a normal space.

Exercise 3b
Prove or disprove:
Suppose that X_\alpha is compact for all but one \alpha \in W such that the non-compact factor is a normal space. Then the product space \prod_{\alpha \in W} \ X_\alpha is a normal space.

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Exercise 4

Exercise 4a
Let X be a compact space.
Show that if X^n is hereditarily Lindelof for all positive integer n, then X is metrizable.

Exercise 4b
Prove or disprove:
If X^n is hereditarily Lindelof for all positive integer n, then X is metrizable.

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Exercise 5

Let Y the product of uncountably many copies of the real line \mathbb{R}. If a specific example is desired, try Y=\mathbb{R}^{\omega_1} (\omega_1 many copies of \mathbb{R}) or Y=\mathbb{R}^{\mathbb{R}} (continuum many copies of \mathbb{R}). It is also OK to use a larger number of copies of the real line.

Note that the space Y is not normal (see here).

Exercise 5a
Since the product space Y is not normal, it is not Lindelof. As an exercise, find an open cover of Y that proves that Y is not Lindelof, i.e. an open cover \mathcal{U} of Y such that no countable subcollection of \mathcal{U} can cover Y.

Exercise 5b
Show that for every open cover \mathcal{U} of the space Y, there is a countable \mathcal{V} \subset \mathcal{U} of Y such that \overline{\mathcal{V}}=Y, i.e. \cup \mathcal{V} is dense in Y. Note that with this property, the space Y is said to be weakly Lindelof.

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Exercise 6

This exercise is about the product Y=\mathbb{R}^{\mathbb{R}} (continuum many copies of \mathbb{R}). Show the following.

  1. Show that Y is separable by exhibiting a countable dense set.
  2. Show that Y is not hereditarily separable by exhibiting a non-separable subspace.
  3. Show that the space Y has a closed and discrete subspace of cardinality continuum.
  4. Show that Y is not first countable.
  5. Show that Y is not a Frechet space.
  6. Show that Y is not a countably tight space.

See here for the definition of Frechet space.

See here for the definition of countably tight space.

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Exercise 7

Consider the product space Y=\mathbb{\omega}^{\omega_1}. It is not normal (see here).

Exercise 7a
Construct a dense normal subspace of Y.

Exercise 7b
Construct a dense Lindelof subspace of Y.

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\copyright \ 2016 \text{ by Dan Ma}