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 . There are four examples of spaces that are CCC but not separable found in  – 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.
The countable chain conditon
A topological space is said to have the countable chain condition (to have the CCC for short) if is a disjoint collection of non-empty open subsets of (meaning that for any with , we have ), then 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 has the CCC, we also say that is a CCC space or X is CCC. A space is separable if there exists a countable subset of such that is dense in (meaning that if is a nonempty open subset of , then ).
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.
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.
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 is the cardinality of continuum, then consider , the product of many copies of , or consider where is your favorite separable space.
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 -products. To make this clear, let’s focus on a specific example. Consider where is the cardinality of continuum. Consider the following subspace.
The subspace is dense in , thus has CCC. It is straightforward to verify that is not separable.
To implement this example, find any space 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 -product, which is the subspace consisting of all points that are non-zero at only countably many coordinates. The -product has the countable chain condition since it is a dense subspace of the CCC space . The -product is not separable since there are uncountably many factors in the product space and that each factor has at least two points. This idea had been implemented in this previous post.
The third class of spaces is the class of Pixley-Roy spaces, which are hyperspaces. Given a space , let be the set of all non-empty finite subsets of . For and for any open subset of , let . The sets over all and form a base for a topology on . This topology is called the Pixley-Roy topology (or Pixley-Roy hyperspace topology). The set with this topology is called a Pixley-Roy space.
The Pixley-Roy hyperspaces are discussed in this previous post. Whenever the ground space is uncountable, is not a separable space. We need to identify the that are CCC. According to the previous post on Pixley-Roy hyperspaces, for any space with a countable network, is CCC. Thus for any uncountable space with a countable network, the Pixley-Roy space is a CCC space that is not separable. The following gives a few such examples.
where is any uncountable, separable and metrizable space.
where 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 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.
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 . Let be any uncountable completely regular space. The product space always has the CCC since it is a product of separable space. Now we single out a dense subspace of for which there is a characterization for separability, namely the subspace , which is the set of all continuous functions from into . The subspace as a topological space is usually denoted by . For a basic discussion of , see this previous post.
We know precisely when is separable. The following theorem captures the idea.
Theorem 1 – Theorem I.1.3 
The function space is separable if and only if the domain space has a weaker (or coarser) separable metric topology (in other words, is submetrizable with a separable metric topology).
Based on the theorem, is separable for any separable metric space . Other examples of separable include spaces 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 would be separable if 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 that have no weaker separable metric topology.
- Any compact space that is not metrizable.
- The space , the first uncountable ordinal with the order topology.
- Any space where is not separable.
The function space 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 are: the first uncountable successor ordinal , the double arrow space, and the product space .
It can be shown that is not separable (see this previous post). The last example is due to the following theorem.
Theorem 2 – Theorem I.1.4 
The function space has a weaker (or coarser) separable metric topology if and only if the domain space is separable.
Thus picking a non-separable space would guarantee that has no weaker separable metric topology. As a result, 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.
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 that is CCC and not separable from any one of the avenues discussed here, the function space has no weaker separable metric topology (by Theorem 2) and the function space 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.
- Show that in metrizable spaces, CCC and separable are equivalent. The only part that needs to be shown is that if is metrizable and CCC, then is separable.
- Show that any dense subspace of a CCC space is also CCC.
- Verify that the space defined in Example 2 is dense in and is not separable.
- Verify that the Pixley-Roy space defined in Example 3 is CCC and not separable.
- Verify that function space mentioned in Example 4 is not separable. Hint: use the pressing down lemma.
- Arkhangelskii, A. V., Topological Function Spaces, Mathematics and Its Applications Series, Kluwer Academic Publishers, Dordrecht, 1992.
- Steen, L. A., Seebach, J. A., Counterexamples in Topology, Dover Publications, Inc., New York, 1995.
Revised October 26, 2018