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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]]).
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1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p0713102.png" /> is paracompact.
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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.
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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]]).
  
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" />.
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1) $  X $
 +
is paracompact.
  
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).
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2) Each open covering of $  X $
 +
can be refined to a locally finite open covering.
  
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" />.
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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 $.
  
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.
+
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).
  
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" />).
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5) For any open covering $  \gamma $
 +
of $  X $
 +
there exists an open covering which is a star refinement of $  \gamma $.
  
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" />.
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6) Each open covering of $  X $
 +
can be refined to a conservative covering.
  
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.
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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====
 
====Comments====
 
Additional equivalent statements are:
 
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]]).
+
10) The product of $  X $
 +
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.
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11) $  X \times \beta X $
 +
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.
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12) Every lower semi-continuous [[Multi-valued mapping|multi-valued mapping]] from $  X $
 +
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.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071310/p07131037.png" />.
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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.
 
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" />.
+
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]].
 
See also [[Paracompact space|Paracompact space]].

Latest revision as of 08:05, 6 June 2020


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
How to Cite This Entry:
Paracompactness criteria. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Paracompactness_criteria&oldid=48112
This article was adapted from an original article by A.V. Arkhangel'skii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article