Editing Paracompact space (section)

== Relationship with compactness ==

There is a similarity between the definitions of [[compact space|compactness]] and paracompactness:
For paracompactness, "subcover" is replaced by "open refinement" and "finite" by is replaced by "locally finite". Both of these changes are significant: if we take the definition of paracompact and change "open refinement" back to "subcover", or "locally finite" back to "finite", we end up with the compact spaces in both cases.

Paracompactness has little to do with the notion of compactness, but rather more to do with breaking up topological space entities into manageable pieces.

=== Comparison of properties with compactness ===

Paracompactness is similar to compactness in the following respects:

* Every closed subset of a paracompact space is paracompact.
* Every paracompact [[Hausdorff space]] is [[normal space|normal]].{{sfn | Dugundji | 1966 | pp=165, Theorem 2.2}}

It is different in these respects:

* A paracompact subset of a Hausdorff space need not be closed. In fact, for metric spaces, all subsets are paracompact.
* A product of paracompact spaces need not be paracompact. The [[Sorgenfrey plane|square of the real line '''R''' in the lower limit topology]] is a classical example for this.