# Definition:Closed Mapping

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## Definition

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.

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction: Functions