Definition:Closed Mapping

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Let $X, Y$ be topological spaces.

Let $f: X \to Y$ be a mapping.

If, for any closed set $V \subseteq X$, the image $\map f V$ is closed in $Y$, then $f$ is referred to as a closed mapping.

Also defined as

The term closed mapping (on a set or class) can also be seen in mapping theory to refer to a mapping whose image is a subset of its preimage:

$f \sqbrk S \subseteq S$

In $\mathsf{Pr} \infty \mathsf{fWiki}$ the preferred way to refer to such a mapping is to apply the term closed to the subset $S$ as being closed under $f$.

Also see

  • Results about closed mappings can be found here.