next up previous contents index
Next: semi-locally connected Up: Definitions Previous: semi-continuum

semi-continuous

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

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

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.

Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih
2001-02-21