# Paracompactness criteria

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

1) $X$ is paracompact.

2) Each open covering of $X$ can be refined to a locally finite open covering.

3) Each open covering of $X$ can be refined to a $\sigma$- locally finite open covering, i.e. an open covering decomposing into a countable collection of locally finite families of sets in $X$.

4) Each open covering of $X$ can be refined to a locally finite covering (about the structure of the elements of which nothing is assumed).

5) For any open covering $\gamma$ of $X$ there exists an open covering which is a star refinement of $\gamma$.

6) Each open covering of $X$ can be refined to a conservative covering.

7) For any open covering $\gamma$ of $X$ there exists a countable collection $\lambda _ {1} , \lambda _ {2} \dots$ of open coverings of this space such that for each point $x \in X$ and for each of its neighbourhoods $O _ {x}$ there exist a $U \in \gamma$ and an integer $i$ satisfying the condition: Each element of $\lambda _ {i}$ intersecting $O _ {x}$ is contained in $U$( i.e. each star of the set $O _ {x}$ relative to $\lambda _ {i}$ lies in $U$).

8) For any open covering $\omega$ of $X$ there exists a continuous mapping of the space $X$ into some metric space $Y$ subject to the condition: At each point of $Y$ there exists a neighbourhood whose inverse image is contained in an element of $\omega$.

9) The space $X$ is collectionwise normal and weakly paracompact.

10) The product of $X$ and any compact Hausdorff space is normal (cf. Normal space).

11) $X \times \beta X$ is normal.

12) Every lower semi-continuous multi-valued mapping from $X$ to a Banach space contains a continuous single-valued mapping.

13) $X$ 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 $\omega$.

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 $\gamma$, in particular a covering, is called a conservative family of sets if for every subfamily $\gamma ^ \prime$ of $\gamma$, $[ \cup _ {P \in \gamma ^ \prime } P] = \cup _ {P \in \gamma ^ \prime } [ P]$. Here $[ A]$ denotes the closure of $A \subset X$.