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A mapping of one topological space to another, under which the image of every closed set is a closed set. The class of continuous closed mappings plays an important role in general topology and its applications. Continuous closed compact mappings are called perfect mappings. A continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022560/c0225601.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022560/c0225602.png" />, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022560/c0225603.png" />-spaces is closed if and only if the decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022560/c0225604.png" /> is continuous in the sense of Aleksandrov (upper continuous) or if for every open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022560/c0225605.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022560/c0225606.png" />, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022560/c0225607.png" /> is open in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022560/c0225608.png" />. The latter property is basic to the definition of upper semi-continuous many-valued mappings. That is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022560/c0225609.png" /> is closed if and only if its (many-valued) inverse mapping is upper continuous. Any continuous mapping of a Hausdorff compactum onto a Hausdorff space is closed. Any continuous closed mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022560/c02256010.png" />-spaces is a quotient mapping; the converse is false. The orthogonal projection of a plane onto a straight line is continuous and open, but not closed. Similarly, not every continuous closed mapping is open. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022560/c02256011.png" /> is continuous and closed, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022560/c02256012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022560/c02256013.png" /> completely regular, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022560/c02256014.png" /> for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022560/c02256015.png" />. (Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022560/c02256016.png" /> is the [[Stone–Čech compactification|Stone–Čech compactification]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022560/c02256017.png" /> is the continuous extension of the mapping to the Stone–Čech compactifications of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022560/c02256018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022560/c02256019.png" />); the converse is true in the class of normal spaces. Passage to the image under a continuous closed mapping preserves the following topological properties: normality; collection-wise normality; perfect normality; paracompactness; weak paracompactness. Complete regularity and strong paracompactness need not be preserved under continuous closed — and even perfect — mappings. Passage to the pre-image under a continuous closed mapping need not preserve the above-mentioned properties. The explanation for this is that the pre-image of a point under a continuous closed mapping need not be compact, though in many cases there is little difference between continuous closed and perfect mappings. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022560/c02256020.png" /> is a continuous closed mapping of a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022560/c02256021.png" /> onto a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022560/c02256022.png" /> satisfying the first axiom of countability, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022560/c02256023.png" /> is metrizable and the boundary of the pre-image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022560/c02256024.png" /> is compact for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022560/c02256025.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022560/c02256026.png" /> is a continuous closed mapping of a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022560/c02256027.png" /> onto a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022560/c02256028.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022560/c02256029.png" />, then the set of all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022560/c02256030.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022560/c02256031.png" /> is not compact is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022560/c02256032.png" />-discrete.
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A mapping of one topological space to another, under which the image of every closed set is a closed set. The class of continuous closed mappings plays an important role in general topology and its applications. Continuous closed compact mappings are called perfect mappings. A continuous mapping $  f : X \rightarrow Y $,  
 +
$  f ( X ) = Y $,  
 +
of $  T _ {1} $-
 +
spaces is closed if and only if the decomposition $  \{ {f ^ { - 1 } y } : {y \in Y } \} $
 +
is continuous in the sense of Aleksandrov (upper continuous) or if for every open set $  U $
 +
in $  X $,  
 +
the set $  f ^ { \# } = \{ {y \in Y } : {f ^ { - 1 } y \in U } \} $
 +
is open in $  U $.  
 +
The latter property is basic to the definition of upper semi-continuous many-valued mappings. That is, $  f $
 +
is closed if and only if its (many-valued) inverse mapping is upper continuous. Any continuous mapping of a Hausdorff compactum onto a Hausdorff space is closed. Any continuous closed mapping of $  T _ {1} $-
 +
spaces is a quotient mapping; the converse is false. The orthogonal projection of a plane onto a straight line is continuous and open, but not closed. Similarly, not every continuous closed mapping is open. If $  f : X \rightarrow Y $
 +
is continuous and closed, with $  X $
 +
and $  Y $
 +
completely regular, then $  \overline{f}\; {} ^ { - 1 } y = [ f ^ { - 1 } y ] \beta X $
 +
for any point $  y \in Y $.  
 +
(Here $  \beta X $
 +
is the [[Stone–Čech compactification|Stone–Čech compactification]] and $  \overline{f}\; : \beta X \rightarrow \beta Y $
 +
is the continuous extension of the mapping to the Stone–Čech compactifications of $  X $
 +
and $  Y $);  
 +
the converse is true in the class of normal spaces. Passage to the image under a continuous closed mapping preserves the following topological properties: normality; collection-wise normality; perfect normality; paracompactness; weak paracompactness. Complete regularity and strong paracompactness need not be preserved under continuous closed — and even perfect — mappings. Passage to the pre-image under a continuous closed mapping need not preserve the above-mentioned properties. The explanation for this is that the pre-image of a point under a continuous closed mapping need not be compact, though in many cases there is little difference between continuous closed and perfect mappings. If $  f $
 +
is a continuous closed mapping of a metric space $  X $
 +
onto a space $  Y $
 +
satisfying the first axiom of countability, then $  Y $
 +
is metrizable and the boundary of the pre-image $  f ^ { - 1 } y $
 +
is compact for every $  y \in Y $.  
 +
If $  f $
 +
is a continuous closed mapping of a metric space $  X $
 +
onto a $  T _ {1} $-
 +
space $  Y $,  
 +
then the set of all points $  y \in Y $
 +
for which $  f ^ { - 1 } y $
 +
is not compact is $  \sigma $-
 +
discrete.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.V. Arkhangel'skii,  "Mappings and spaces"  ''Russian Math. Surveys'' , '''21''' :  4  (1966)  pp. 115–126  ''Uspekhi Mat. Nauk'' , '''21''' :  4  (1966)  pp. 133–184</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.V. Arkhangel'skii,  V.I. Ponomarev,  "Fundamentals of general topology: problems and exercises" , Reidel  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R. Engelking,  "General topology" , PWN  (1977)  (Translated from Polish)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.V. Arkhangel'skii,  "Mappings and spaces"  ''Russian Math. Surveys'' , '''21''' :  4  (1966)  pp. 115–126  ''Uspekhi Mat. Nauk'' , '''21''' :  4  (1966)  pp. 133–184</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.V. Arkhangel'skii,  V.I. Ponomarev,  "Fundamentals of general topology: problems and exercises" , Reidel  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R. Engelking,  "General topology" , PWN  (1977)  (Translated from Polish)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The notion of a closed mapping leads to the notion of an upper semi-continuous decomposition of a space. This is a decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022560/c02256033.png" /> of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022560/c02256034.png" /> such that the quotient mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022560/c02256035.png" /> is closed.
+
The notion of a closed mapping leads to the notion of an upper semi-continuous decomposition of a space. This is a decomposition $  E $
 +
of a space $  X $
 +
such that the quotient mapping $  q: X \rightarrow X/E $
 +
is closed.
  
In the Russian literature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022560/c02256036.png" /> denotes the closure of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022560/c02256037.png" />, so in this article <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022560/c02256038.png" /> is the closure of the fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022560/c02256039.png" /> in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022560/c02256040.png" /> (see also [[Closure of a set|Closure of a set]]).
+
In the Russian literature $  [ A] $
 +
denotes the closure of the set $  A $,  
 +
so in this article $  [ f ^ { - 1 } y] \beta X $
 +
is the closure of the fibre $  f ^ { - 1 } y $
 +
in the space $  \beta X $(
 +
see also [[Closure of a set|Closure of a set]]).

Latest revision as of 17:44, 4 June 2020


A mapping of one topological space to another, under which the image of every closed set is a closed set. The class of continuous closed mappings plays an important role in general topology and its applications. Continuous closed compact mappings are called perfect mappings. A continuous mapping $ f : X \rightarrow Y $, $ f ( X ) = Y $, of $ T _ {1} $- spaces is closed if and only if the decomposition $ \{ {f ^ { - 1 } y } : {y \in Y } \} $ is continuous in the sense of Aleksandrov (upper continuous) or if for every open set $ U $ in $ X $, the set $ f ^ { \# } = \{ {y \in Y } : {f ^ { - 1 } y \in U } \} $ is open in $ U $. The latter property is basic to the definition of upper semi-continuous many-valued mappings. That is, $ f $ is closed if and only if its (many-valued) inverse mapping is upper continuous. Any continuous mapping of a Hausdorff compactum onto a Hausdorff space is closed. Any continuous closed mapping of $ T _ {1} $- spaces is a quotient mapping; the converse is false. The orthogonal projection of a plane onto a straight line is continuous and open, but not closed. Similarly, not every continuous closed mapping is open. If $ f : X \rightarrow Y $ is continuous and closed, with $ X $ and $ Y $ completely regular, then $ \overline{f}\; {} ^ { - 1 } y = [ f ^ { - 1 } y ] \beta X $ for any point $ y \in Y $. (Here $ \beta X $ is the Stone–Čech compactification and $ \overline{f}\; : \beta X \rightarrow \beta Y $ is the continuous extension of the mapping to the Stone–Čech compactifications of $ X $ and $ Y $); the converse is true in the class of normal spaces. Passage to the image under a continuous closed mapping preserves the following topological properties: normality; collection-wise normality; perfect normality; paracompactness; weak paracompactness. Complete regularity and strong paracompactness need not be preserved under continuous closed — and even perfect — mappings. Passage to the pre-image under a continuous closed mapping need not preserve the above-mentioned properties. The explanation for this is that the pre-image of a point under a continuous closed mapping need not be compact, though in many cases there is little difference between continuous closed and perfect mappings. If $ f $ is a continuous closed mapping of a metric space $ X $ onto a space $ Y $ satisfying the first axiom of countability, then $ Y $ is metrizable and the boundary of the pre-image $ f ^ { - 1 } y $ is compact for every $ y \in Y $. If $ f $ is a continuous closed mapping of a metric space $ X $ onto a $ T _ {1} $- space $ Y $, then the set of all points $ y \in Y $ for which $ f ^ { - 1 } y $ is not compact is $ \sigma $- discrete.

References

[1] A.V. Arkhangel'skii, "Mappings and spaces" Russian Math. Surveys , 21 : 4 (1966) pp. 115–126 Uspekhi Mat. Nauk , 21 : 4 (1966) pp. 133–184
[2] A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)
[3] R. Engelking, "General topology" , PWN (1977) (Translated from Polish)

Comments

The notion of a closed mapping leads to the notion of an upper semi-continuous decomposition of a space. This is a decomposition $ E $ of a space $ X $ such that the quotient mapping $ q: X \rightarrow X/E $ is closed.

In the Russian literature $ [ A] $ denotes the closure of the set $ A $, so in this article $ [ f ^ { - 1 } y] \beta X $ is the closure of the fibre $ f ^ { - 1 } y $ in the space $ \beta X $( see also Closure of a set).

How to Cite This Entry:
Closed mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Closed_mapping&oldid=46364
This article was adapted from an original article by V.I. Ponomarev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article