# Semi-continuous mapping

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).

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.