# Paracompactness is weakly hereditary

From Topospaces

This article gives the statement, and possibly proof, of a topological space property (i.e., paracompact space) satisfying a topological space metaproperty (i.e., weakly hereditary property of topological spaces)

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## Statement

### Property-theoretic statement

The property of being a paracompact space is a weakly hereditary property of topological spaces.

### Verbal statement

Any closed subset of a paracompact space is a paracompact space with the subspace topology.

## Related facts

## Proof

**Given**: A paracompact space , a closed subset of .

**To prove**: Consider an open cover of by open sets with , an indexing set. The have a locally finite open refinement.

**Proof**:

- By the definition of subspace topology, we can find open sets of such that , thus the union of the s contains .
- Since is closed, we can
*throw in*the open set , and get an open cover of the*whole space*. - Since the whole space is compact, this open cover has a locally finite open refinement. In other words, there is a locally finite open refinement of the s, that, possibly along with , covers the whole of .
- By
*throwing out*any member of this new refinement that do not intersect , we get a locally finite open refinement of s whose union contains . The intersections of these with form a locally finite open refinement of the s: The main point here is that if an open set in the refinement is contained in , its intersection with is contained in .