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''upper (lower)''
 
''upper (lower)''
  
A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084020/s0840201.png" /> from a [[Topological space|topological space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084020/s0840202.png" /> into a partially ordered set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084020/s0840203.png" /> such that
+
A mapping $  f $
 +
from a [[Topological space|topological space]] $  X $
 +
into a partially ordered set $  P $
 +
such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084020/s0840204.png" /></td> </tr></table>
+
$$
 +
\lim\limits  x _ {n}  = x
 +
$$
  
 
implies that
 
implies that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084020/s0840205.png" /></td> </tr></table>
+
$$
 
+
\overline{\lim\limits}\; f ( x _ {n} )  \leq  f ( x) \  \left ( fnnme \underline{lim}  f ( x _ {n} )
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084020/s0840206.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084020/s0840207.png" />) denotes the limes superior (inferior).
+
\geq  f ( x) \right ) ,
 
+
$$
  
 +
where  $  \overline{\lim\limits}\; $(
 +
$  fnnme \underline{lim} $)
 +
denotes the [[limes superior]] (inferior).
  
 
====Comments====
 
====Comments====
On a partially ordered set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084020/s0840208.png" /> the collection consisting of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084020/s0840209.png" /> and all sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084020/s08402010.png" /> is a base for a topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084020/s08402011.png" />, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084020/s08402012.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084020/s08402013.png" /> and all sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084020/s08402014.png" /> define a topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084020/s08402015.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084020/s08402016.png" /> is upper semi-continuous, (u.s.c.) (respectively, lower semi-continuous (l.s.c.)) if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084020/s08402017.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084020/s08402018.png" />) is continuous.
+
On a partially ordered set $  P $
 +
the collection consisting of $  P $
 +
and all sets $  U _ {x} = \{ {y \in P } : {y < x } \} $
 +
is a base for a topology on $  P $,  
 +
denoted by $  \tau _ {-} $,  
 +
and $  P $
 +
and all sets $  V _ {x} = \{ {y \in P } : {y \geq  x } \} $
 +
define a topology $  \tau _ {+} $.  
 +
The mapping $  f: X \rightarrow P $
 +
is upper semi-continuous, (u.s.c.) (respectively, lower semi-continuous (l.s.c.)) if and only if $  f : X \rightarrow ( P , \tau _ {+} ) $(
 +
respectively, $  f: X \rightarrow ( P , \tau _ {-} ) $)  
 +
is continuous.
  
In fact, upper and lower semi-continuity are usually defined only for mappings to the real line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084020/s08402019.png" />. In terms of open sets, one sees that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084020/s08402020.png" /> is upper (lower) semi-continuous if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084020/s08402021.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084020/s08402022.png" />) is open for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084020/s08402023.png" />.
+
In fact, upper and lower semi-continuity are usually defined only for mappings to the real line $  \mathbf R $.  
 +
In terms of open sets, one sees that $  f : X \rightarrow \mathbf R $
 +
is upper (lower) semi-continuous if and only if $  f ^ { - 1 } [ (- \infty , a)] $(
 +
$  f ^ { - 1 } [( a, \infty )] $)  
 +
is open for every $  a $.
  
Semi-continuity is also defined for set-valued mappings. A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084020/s08402024.png" /> is upper (lower) semi-continuous if for every open subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084020/s08402025.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084020/s08402026.png" /> the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084020/s08402027.png" /> (the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084020/s08402028.png" />) is open.
+
Semi-continuity is also defined for set-valued mappings. A mapping $  F : X \rightarrow 2  ^ {Y} $
 +
is upper (lower) semi-continuous if for every open subset $  U $
 +
of $  Y $
 +
the set $  \{ {x } : {F( x) \subseteq U } \} $(
 +
the set $  \{ {x } : {F( x) \cap U \neq \emptyset } \} $)  
 +
is open.
  
Note that if a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084020/s08402029.png" /> is regarded as a set-valued mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084020/s08402030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084020/s08402031.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084020/s08402032.png" /> is lower semi-continuous if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084020/s08402033.png" /> is lower semi-continuous; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084020/s08402034.png" /> is upper semi-continuous if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084020/s08402035.png" /> is upper semi-continuous.
+
Note that if a mapping $  f: X \rightarrow Y $
 +
is regarded as a set-valued mapping $  F : X \rightarrow 2  ^ {Y} $,  
 +
$  F( x) = \{ f( x) \} $,  
 +
then $  F $
 +
is lower semi-continuous if and only if $  f $
 +
is lower semi-continuous; and $  F $
 +
is upper semi-continuous if and only if $  f $
 +
is upper semi-continuous.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Engelking,  "General topology" , Heldermann  (1989)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Engelking,  "General topology" , Heldermann  (1989)</TD></TR></table>

Latest revision as of 08:13, 6 June 2020


upper (lower)

A mapping $ f $ from a topological space $ X $ into a partially ordered set $ P $ such that

$$ \lim\limits x _ {n} = x $$

implies that

$$ \overline{\lim\limits}\; f ( x _ {n} ) \leq f ( x) \ \left ( fnnme \underline{lim} f ( x _ {n} ) \geq f ( x) \right ) , $$

where $ \overline{\lim\limits}\; $( $ fnnme \underline{lim} $) denotes the limes superior (inferior).

Comments

On a partially ordered set $ P $ the collection consisting of $ P $ and all sets $ U _ {x} = \{ {y \in P } : {y < x } \} $ is a base for a topology on $ P $, denoted by $ \tau _ {-} $, and $ P $ and all sets $ V _ {x} = \{ {y \in P } : {y \geq x } \} $ define a topology $ \tau _ {+} $. The mapping $ f: X \rightarrow P $ is upper semi-continuous, (u.s.c.) (respectively, lower semi-continuous (l.s.c.)) if and only if $ f : X \rightarrow ( P , \tau _ {+} ) $( respectively, $ f: X \rightarrow ( P , \tau _ {-} ) $) is continuous.

In fact, upper and lower semi-continuity are usually defined only for mappings to the real line $ \mathbf R $. In terms of open sets, one sees that $ f : X \rightarrow \mathbf R $ is upper (lower) semi-continuous if and only if $ f ^ { - 1 } [ (- \infty , a)] $( $ f ^ { - 1 } [( a, \infty )] $) is open for every $ a $.

Semi-continuity is also defined for set-valued mappings. A mapping $ F : X \rightarrow 2 ^ {Y} $ is upper (lower) semi-continuous if for every open subset $ U $ of $ Y $ the set $ \{ {x } : {F( x) \subseteq U } \} $( the set $ \{ {x } : {F( x) \cap U \neq \emptyset } \} $) is open.

Note that if a mapping $ f: X \rightarrow Y $ is regarded as a set-valued mapping $ F : X \rightarrow 2 ^ {Y} $, $ F( x) = \{ f( x) \} $, then $ F $ is lower semi-continuous if and only if $ f $ is lower semi-continuous; and $ F $ is upper semi-continuous if and only if $ f $ is upper semi-continuous.

References

[a1] R. Engelking, "General topology" , Heldermann (1989)
How to Cite This Entry:
Semi-continuous mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-continuous_mapping&oldid=13587
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article