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.

Comments

Additional equivalent statements are:

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 $.

See also Paracompact space.

References

[a1]	D.K. Burke, "Covering properties" K. Kunen (ed.) J.E. Vaughan (ed.) , Handbook of Set-Theoretic Topology , North-Holland (1984) pp. Chapt. 9; pp. 347–422
[a2]	E.A. Michael, "A note on paracompact spaces" Proc. Amer. Math. Soc. , 4 (1953) pp. 831–838
[a3]	E.A. Michael, "Another note on paracompact spaces" Proc. Amer. Math. Soc. , 8 (1958) pp. 822–828
[a4]	E.A. Michael, "Yet another note on paracompact spaces" Proc. Amer. Math. Soc. , 10 (1959) pp. 309–314
[a5]	A.H. Stone, "Paracompactness and product spaces" Bull. Amer. Math. Soc. , 54 (1948) pp. 977–982
[a6]	J. Isbell, "Supercomplete spaces" Pacific J. Math. , 12 (1962) pp. 287–290
[a7]	E. Michael, "Continuous selections I" Ann. of Math. (2) , 63 (1956) pp. 361–382
[a8]	H. Tamano, "On paracompactness" Pacific J. Math. , 10 (1960) pp. 1043–1047