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''of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085800/s0858001.png" /> under a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085800/s0858002.png" />''
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$#C+1 = 12 : ~/encyclopedia/old_files/data/S085/S.0805800 Small image
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The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085800/s0858003.png" /> of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085800/s0858004.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085800/s0858005.png" />. An equivalent definition is: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085800/s0858006.png" />. Closed and irreducible mappings may be characterized by means of small images. A [[Continuous mapping|continuous mapping]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085800/s0858007.png" /> is closed (cf. [[Closed mapping|Closed mapping]]) if and only if the small image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085800/s0858008.png" /> of each open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085800/s0858009.png" /> is open. A continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085800/s08580010.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085800/s08580011.png" /> is closed and irreducible (cf. [[Irreducible mapping|Irreducible mapping]]) if and only if the small image of each non-empty open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085800/s08580012.png" /> is a non-empty set.
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''of a set  $  A \subset  X $
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under a mapping  $  f:  X \rightarrow Y $''
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The set $  f ^ { \srp } A $
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of all $  y \in Y $
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for which $  f ^ { - 1 } y \subset  A $.  
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An equivalent definition is: $  f ^ { \srp } A = Y \setminus  f ( X \setminus  A) $.  
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Closed and irreducible mappings may be characterized by means of small images. A [[Continuous mapping|continuous mapping]] $  f: X \rightarrow Y $
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is closed (cf. [[Closed mapping|Closed mapping]]) if and only if the small image $  f ^ { \srp } U $
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of each open set $  U \subset  X $
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is open. A continuous mapping $  f: X \rightarrow Y $
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onto $  Y $
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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 $
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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
This article was adapted from an original article by V.V. Fedorchuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article