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of a set $ A \subset X $ under a mapping $ f: X \rightarrow Y $

The set $ f ^ { \sharp } A $ of all $ y \in Y $ for which the fibre $ f ^ { - 1 } y \subset A $. An equivalent definition is: $ f ^ { \sharp } A = Y \setminus f ( X \setminus A) $. Closed and irreducible mappings may be characterized by means of small images. A continuous mapping $ f: X \rightarrow Y $ is closed (cf. Closed mapping) if and only if the small image $ f ^ { \sharp } U $ of each open set $ U \subset X $ is open. A continuous mapping $ f: X \rightarrow Y $ onto $ Y $ is closed and irreducible (cf. Irreducible mapping) if and only if the small image of each non-empty open set $ U \subset X $ is a non-empty set.

How to Cite This Entry:
Small image. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Small_image&oldid=51638
This article was adapted from an original article by V.V. Fedorchuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article