# The category of paracompact spaces

### And all its properties

### Heredity along constructions

One drawback of the notion of paracompactness is that it is not as robust as the notion of compactness along usual constructions. It is also less preserved than regularity and Lindelöfness.

- A product of paracompact spaces need
*not*be paracompact; - A closed subspace of a paracompact space is paracompact;
- A filtered colimit of paracompact spaces need
*not*be paracompact; - A disjoint sum of paracompact spaces is paracompact;