View source for Paracompactness criteria

The following statements are equivalent for an arbitrary completely-regular Hausdorff space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p0713101.png" /> (cf. [[Completely-regular space|Completely-regular space]]; [[Hausdorff space|Hausdorff space]]).

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p0713102.png" /> is paracompact.

2) Each open covering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p0713103.png" /> can be refined to a locally finite open covering.

3) Each open covering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p0713104.png" /> can be refined to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p0713106.png" />-locally finite open covering, i.e. an open covering decomposing into a countable collection of locally finite families of sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p0713107.png" />.

4) Each open covering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p0713108.png" /> can be refined to a locally finite covering (about the structure of the elements of which nothing is assumed).

5) For any open covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p0713109.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p07131010.png" /> there exists an open covering which is a star refinement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p07131011.png" />.

6) Each open covering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p07131012.png" /> can be refined to a conservative covering.

7) For any open covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p07131013.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p07131014.png" /> there exists a countable collection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p07131015.png" /> of open coverings of this space such that for each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p07131016.png" /> and for each of its neighbourhoods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p07131017.png" /> there exist a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p07131018.png" /> and an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p07131019.png" /> satisfying the condition: Each element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p07131020.png" /> intersecting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p07131021.png" /> is contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p07131022.png" /> (i.e. each star of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p07131023.png" /> relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p07131024.png" /> lies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p07131025.png" />).

8) For any open covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p07131026.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p07131027.png" /> there exists a continuous mapping of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p07131028.png" /> into some metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p07131029.png" /> subject to the condition: At each point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p07131030.png" /> there exists a neighbourhood whose inverse image is contained in an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p07131031.png" />.

9) The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p07131032.png" /> is collectionwise normal and weakly paracompact.



====Comments====
Additional equivalent statements are:

10) The product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p07131033.png" /> and any compact Hausdorff space is normal (cf. [[Normal space|Normal space]]).

11) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p07131034.png" /> is normal.

12) Every lower semi-continuous [[Multi-valued mapping|multi-valued mapping]] from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p07131035.png" /> to a Banach space contains a continuous single-valued mapping.

13) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p07131036.png" /> admits a uniformity for which the [[Hyperspace|hyperspace]] of closed sets is complete.

Such a mapping as is posited in 8) is said to realize the covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p07131037.png" />.

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p07131038.png" />, in particular a covering, is called a conservative family of sets if for every subfamily <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p07131039.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p07131040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p07131041.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p07131042.png" /> denotes the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p07131043.png" />.

See also [[Paracompact space|Paracompact space]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.A. Michael,   "A note on paracompact spaces"  ''Proc. Amer. Math. Soc.'' , '''4'''  (1953)  pp. 831–838</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E.A. Michael,   "Another note on paracompact spaces"  ''Proc. Amer. Math. Soc.'' , '''8'''  (1958)  pp. 822–828</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E.A. Michael,   "Yet another note on paracompact spaces"  ''Proc. Amer. Math. Soc.'' , '''10'''  (1959)  pp. 309–314</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A.H. Stone,   "Paracompactness and product spaces"  ''Bull. Amer. Math. Soc.'' , '''54'''  (1948)  pp. 977–982</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J. Isbell,   "Supercomplete spaces"  ''Pacific J. Math.'' , '''12'''  (1962)  pp. 287–290</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  E. Michael,   "Continuous selections I"  ''Ann. of Math. (2)'' , '''63'''  (1956)  pp. 361–382</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  H. Tamano,   "On paracompactness"  ''Pacific J. Math.'' , '''10'''  (1960)  pp. 1043–1047</TD></TR></table>