It is a well known fact in general topology that in the class of spaces with the countable chain condition (ccc), paracompactness equals the Lindelof property. I would like to write down a proof for this fact, proving that paracompact space with the ccc is Lindelof. A space has the countable chain condition (ccc) if every pairwise disjoint family of open sets in the space is countable. I would like to make an observation about two classes of ccc spaces. One is the class of separable spaces. The other is the product of separable spaces. In particular, the product of real lines has the ccc. So has the ccc for any cardinal . The countable chain condition is hereditary with respect to dense subspaces. The function space , the real-valued continuous function space with the pointwise convergence topology, is always a dense subspace of . Thus always has the ccc. For this class of function spaces, there is no distinction between paracompactness and the Lindelof property (if it is paracompact, it is Lindelof).

**Theorem**. If a paracompact space has the ccc, then is paracompact.

This theorem is established after the following lemma is proved.

**Lemma**. In a space with the ccc, any locally finite open cover is countable.

**Proof of Lemma**. Let be a space with the countable chain condition (ccc). Let be a locally finite open cover of . For each , let be open such that meets at most finitely many open sets in . Let .

For , a finite collection is said to be a chain from to if , , and for . For each , let there is a chain from to . Clearly each is a countable collection. For each , let . For , if , then there is a chain from to and there is a chain from to . This means . Consider the distinct elements where . These open sets form a pairwise disjoint collection of open sets in . By the ccc, there are only countably many distinct .

Let be the countable collection of all distinct open sets . Each is associated with countably many . In turn, each for some . There is a one-to-countable mapping from onto . Since is countable, is countable.

### Like this:

Like Loading...

*Related*