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.
Comments
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.
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 |
Paracompactness criteria. A.V. Arkhangel'skii (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Paracompactness_criteria&oldid=12705