Closed map

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Created quickly, just to document the concept

A open map is a thing to and is defined similarly.

Definition

Let $(X,\mathcal{ J })$ and $(Y,\mathcal{ K })$ be topological spaces and let $f:X\rightarrow Y$ be a map (not necessarily continuous - just a map between $X$ and $Y$ considered as sets), then we call $f$ a closed map if:

• $\forall U\in C(X,\mathcal{J})[f(U)\in C(Y,\mathcal{K})]$ - that is, that the images (under $f$) of all closed sets of $(X,\mathcal{ J })$ are closed in $(Y,\mathcal{ K })$
  • $C(X,\mathcal{J})$ denotes the set of all closed sets of the topological space $(X,\mathcal{ J })$

Reformulation

A set is closed if its complement is open, so we could state:

• A mapping $f:X\rightarrow Y$ is a closed map if:
  • $\forall E\in\mathcal{P}(X)[(X-E)\in\mathcal{J}\implies (Y-f(E))\in\mathcal{K}]$ - this is Claim 1

See proof of claims below.

Note: we really have an if and only if relationship here. See definitions and iff for information

Proof of claims

Claim 1: Reformulation

Grade: D

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Easy proof, skipped. Proof would be that a map is closed WRT the definition if and only if it satisfies the reformulation

This proof has been marked as an page requiring an easy proof