Paracompactness is weakly hereditary
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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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.
- 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 .