# Paracompactness criteria

The following statements are equivalent for an arbitrary completely-regular Hausdorff space (cf. Completely-regular space; Hausdorff space).

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.

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 .