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
of subsets of a topological space is said to be closure-preserving if for every subfamily
,
:
.
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
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Theorems in topology