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

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

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

13)  $  X $
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  $  \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|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>