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).

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)