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.