Closed map

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Created quickly, just to document the concept
A open map is a thing to and is defined similarly.


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[1]:

  • [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]


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


  1. Introduction to Topological Manifolds - John M. Lee