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