Difference between revisions of "Small image"
From Encyclopedia of Mathematics
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| + | $#C+1 = 12 : ~/encyclopedia/old_files/data/S085/S.0805800 Small image | ||
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| + | ''of a set $ A \subset X $ | ||
| + | under a mapping $ f: X \rightarrow Y $'' | ||
| + | |||
| + | The set $ f ^ { \srp } A $ | ||
| + | of all $ y \in Y $ | ||
| + | for which $ f ^ { - 1 } y \subset A $. | ||
| + | An equivalent definition is: $ f ^ { \srp } A = Y \setminus f ( X \setminus A) $. | ||
| + | Closed and irreducible mappings may be characterized by means of small images. A [[Continuous mapping|continuous mapping]] $ f: X \rightarrow Y $ | ||
| + | is closed (cf. [[Closed mapping|Closed mapping]]) if and only if the small image $ f ^ { \srp } 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|Irreducible mapping]]) if and only if the small image of each non-empty open set $ U \subset X $ | ||
| + | is a non-empty set. | ||
Revision as of 08:14, 6 June 2020
of a set $ A \subset X $
under a mapping $ f: X \rightarrow Y $
The set $ f ^ { \srp } A $ of all $ y \in Y $ for which $ f ^ { - 1 } y \subset A $. An equivalent definition is: $ f ^ { \srp } 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 ^ { \srp } 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=17273
Small image. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Small_image&oldid=17273
This article was adapted from an original article by V.V. Fedorchuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article