View source for Semi-continuous mapping

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''upper (lower)''

A mapping  $  f $
from a [[Topological space|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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Engelking,   "General topology" , Heldermann  (1989)</TD></TR></table>