The theorem indicated by the title is a basic result and is an interesting theorem. It is known that “compact x normal” needs not be normal (see an example here). But for a compact factor and a paracompact factor, the product is not only normal, it is paracompact. I would like to write down a proof. This theorem will be used in subsequent posts. If the factor is Lindelof instead, it can be shown that is Lindelof.
Let be a compact space and let be a paracompact space. Let be an open cover of . For each , let be a finite cover of .
Claim. Fix . There is an open set such that . For each , let be open and let be open such that , , and . Since is compact, choose finitely many open sets that cover , say . Let . It is clear that .
The collection is an open cover of . Then it has a locally finite open refinement . For each , choose such that . Consider where . Let be the collection of all such open sets .
There are three things to check here. One is that is an open cover of . Fix . Then for some . Note that for some . This means that the point for some in .
The second to check is that is a refinement of the original open cover . This is clear since every set in is chosen to be a subset of some set in .
The third is that is a locally finite collection. To see this, let . There is an open such that and can only meet finitely many . For each such , is associated with finitely many sets in . This means is an open set containing that can meet only finitely many sets in . This completes the proof that is paracompact.
If the factor Y is Lindelof, then we can modify the proof to show that is Lindelof. In the step above where is obtained, we get a countable subcover of . Each member of this countable subcover is associated with a finite . Thus we can obtain a countable subcover of .