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semi-continuous

Let a continuum

, a compact space

and a function $F: X \to 2^Y$$ be given. Put

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

The function

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.

Next: semi-locally connected Up: Definitions Previous: semi-continuum

Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih
2001-11-30