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 and regular. A 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 making a statement or stating a result, if has the CCC, we also say that is a CCC space or is CCC. We prove the following theorem.
Let be a space. Then the following conditions are equivalent.
- The space has the CCC.
- For any collection of non-empty open subsets of , there exists a countable such that .
Proof of Theorem 1
Suppose that condition 2 does not hold. Then there exists a collection of non-empty open subsets of such that for any countable , there exists a point such that . From the collection , performing a transfinite inductive process, we will generate an uncountable collection of pairwise disjoint non-empty open subsets of .
Choose some . Choose some . For , suppose that the following have been chosen
such that for each , and . Then by the assumption about , there exists such that . Now choose some such that . The inductive process is completed.
For each , let . Clearly since . For , we have . With the non-empty open sets being pairwise disjoint, we conclude that does not have the CCC.
This is the easier direction. Suppose is not CCC. Let be a collection of pairwise disjoint non-empty open subsets of . It is clear that for any countable , the closure has to miss some (e.g. choose some ). Thus condition 2 does not hold.
Weakly Lindelof Spaces
With Theorem 1, the CCC property looks like a covering property. Let be a CCC space. Let be an open cover of . By Theorem 1, there is a countable such that (in other words, is dense in ). So any CCC space satisfies the following covering property:
For any open cover of , there exists a countable such that is dense in .
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.