View source for Small image

<!--
s0858001.png
$#A+1 = 12 n = 0
$#C+1 = 12 : ~/encyclopedia/old_files/data/S085/S.0805800 Small image
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

''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.