Small image

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.