semi-continuous

Let a continuum X, a compact space Y and a function $F: X \to 2^Y$ be given. Put

$$F^{-1}(B) = \{x \in X: F(x) \cap B \ne \emptyset\}.$$

The function F is said to be lower (upper) semi-continuous provided that F^-1(B) is open (closed) for each open (closed) subset $B \subset Y$. It is said to be continuous provided that it is both lower and upper semi-continuous. This notion of continuity agrees with the one for mappings between metric spaces.