# Closed map

This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
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[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]

### 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