A space is said to have countable chain condition (ccc) if every disjoint collection of open subsets of is countable. When a space has ccc, for convenience we also say that it is a ccc space. We present some basic results about ccc space as well as an equivalent condition for ccc in terms of relatively locally finite open collection. A corollary of this equivalent condition is that in the class of ccc spaces, paracompactness coincides with the Lindelof property.
Let be a space. We have the following simple results about ccc spaces.
- If is separable, then has ccc.
- If is hereditarily Lindelof, then has ccc.
- The property ccc is hereditary with respect to open subspaces.
- Let be a dense subspace of . Then has ccc if and only if has ccc.
We only prove result 4.
Proof of 4
Let be a dense subspace of the space . Suppose that does not have ccc. Then there is an uncountable disjoint collection of open subsets of . Assuming that is an uncountable set, the collection of all (where ) is an uncountable disjoint collection of open subsets of . It follows that does not have ccc. Thus has ccc implies has ccc.
Suppose that does not have ccc. Then there is an uncountable disjoint collection of open subsets of . For each , there is some , open subset of , such that . Let be the collection of all . Note that is uncountable and is a disjoint collection of open subsets of . It follows that does not have ccc. Thus has ccc implies has ccc.
Theorem 1 and Corollary 2 (stated below) are discussed in a previous post (Product of Spaces with Countable Chain Condition). Theorem 1 implies that if the ccc property is preserved by taking product of any finite number of factors, then the ccc property is preserved by taking product of any number of factors. As a corollary, it follows that the product of any number of separable spaces has ccc.
Suppose that is a family of spaces such that has countable chain condition for every finite . Then has countable chain condition.
Suppose that is a family of separable spaces. Then has countable chain condition.
An Equivalent Condition for Countable Chain Condition
The next theorem is an alternative way of looking at countable chain condition. Corollary 3 is an application of this equivalent condition.
Let be a space. Then the following conditions are equivalent.
- If is a collection of open subsets of such that is locally countable in the open subspace , then is countable.
- If is a collection of open subsets of such that is locally finite in the open subspace , then is countable.
- has ccc.
Note that when an open collection is locally finite (locally countable) in the open subspace , is said to be a relatively locally finite (locally countable) open collection.
The directions and are clear.
Suppose that has ccc. Let be a collection of open subsets of such that is locally countable in the open subspace . For each , choose a non-empty open set such that for at most countably many . Let be the collection of all (over all ). Note that is a countable-to-one mapping from into .
For , a chain from to is any finite collection
such that , , and for , . For each , let be the following:
Note that every meets only countably many sets in . Thus every meets only countably many sets in . As a result, each is countable. For each , let . For , if , , and as a result . Thus the collection of all distinct is a collection of disjoint open sets in . Since has ccc, there can be only countably many .
Each is the union of countably many open sets, namely the open sets in , which is countable. Each set is traced back to at most countably many sets in open sets in the original collection . Since the mapping is a countable-to-one map from to , is countable.
Let be a space with countable chain condition. Then is a paracompact space if and only if is a Lindelof space.
The direction is the theorem that every regular Lindelof space is paracompact. See Theorem 3.8.11 and Theorem 5.1.2 in . Also see Corollary 20.8 in .
This direction is a corollary of Theorem 3. Since is paracompact, every open cover of the space has a locally finite open refinement. By Theorem 3, the locally finite open refinement must be countable.
- Engelking, R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.
- Willard, S., General Topology, Addison-Wesley Publishing Company, 1970.