Open map
From Maths
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Created quickly, just to document the concept
- A closed map is a thing too and is defined similarly.
Contents
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] an open map if[1]:
- [ilmath]\forall U\in\mathcal{J}[f(U)\in\mathcal{K}][/ilmath] - that is, that the images (under [ilmath]f[/ilmath]) of all open sets of [ilmath](X,\mathcal{ J })[/ilmath] are open in [ilmath](Y,\mathcal{ K })[/ilmath]
