1) is paracompact.
2) Each open covering of can be refined to a locally finite open covering.
3) Each open covering of can be refined to a -locally finite open covering, i.e. an open covering decomposing into a countable collection of locally finite families of sets in .
4) Each open covering of can be refined to a locally finite covering (about the structure of the elements of which nothing is assumed).
5) For any open covering of there exists an open covering which is a star refinement of .
6) Each open covering of can be refined to a conservative covering.
7) For any open covering of there exists a countable collection of open coverings of this space such that for each point and for each of its neighbourhoods there exist a and an integer satisfying the condition: Each element of intersecting is contained in (i.e. each star of the set relative to lies in ).
8) For any open covering of there exists a continuous mapping of the space into some metric space subject to the condition: At each point of there exists a neighbourhood whose inverse image is contained in an element of .
9) The space is collectionwise normal and weakly paracompact.
Additional equivalent statements are:
10) The product of and any compact Hausdorff space is normal (cf. Normal space).
11) is normal.
12) Every lower semi-continuous multi-valued mapping from to a Banach space contains a continuous single-valued mapping.
13) admits a uniformity for which the hyperspace of closed sets is complete.
Such a mapping as is posited in 8) is said to realize the covering .
Weakly paracompact spaces are also called metacompact. They are the spaces every open covering of which has a point-finite open refinement.
A family of sets , in particular a covering, is called a conservative family of sets if for every subfamily of , . Here denotes the closure of .
See also Paracompact space.
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Paracompactness criteria. A.V. Arkhangel'skii (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Paracompactness_criteria&oldid=12705