Difference between revisions of "Open map"

Revision as of 19:59, 26 September 2016

A closed 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 $f$ an open map if^[1]:

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

References

1. Jump up ↑ Introduction to Topological Manifolds - John M. Lee