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In mathematics, Michael's theorem gives sufficient conditions for a regular
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
(in fact, for a T1-space) to be
paracompact In mathematics, a paracompact space is a topological space in which every open cover has an open Cover (topology)#Refinement, refinement that is locally finite collection, locally finite. These spaces were introduced by . Every compact space is par ...
.


Statement

A family E_i of subsets of a topological space is said to be closure-preserving if for every subfamily E_, :\overline = \bigcup \overline. For example, a locally finite family of subsets has this property. With this terminology, the theorem states: Frequently, the theorem is stated in the following form: In particular, a regular-Hausdorff Lindelöf space is paracompact. The proof of the theorem uses the following result which does not need regularity:


Proof sketch

The proof of the proposition uses the following general lemma


Notes


References

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Further reading


Michael's Theorem in Ncatlab
{{topology-stub Theorems in topology