Closed map

Stub grade: B

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Created quickly, just to document the concept

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

Definition

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

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

Reformulation

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

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