proof of inverse function theorem (topological spaces)

We only have to prove that whenever AX is an open set, then also B=(f-1)-1(A)=f(A)Y is open (f is an open mapping). Equivalently it is enough to prove that B=YB is closed.

Since f is bijectiveMathworldPlanetmathPlanetmath we have B=YB=f(XA)

As A=XA is closed and since X is compactPlanetmathPlanetmath A is compact too (this and the following are well know properties of compact spaces). Moreover being f continuousPlanetmathPlanetmath we know that also B=f(A) is compact. Finally since Y is HausdorffPlanetmathPlanetmath then B is closed.

Title proof of inverse function theorem (topological spacesMathworldPlanetmath)
Canonical name ProofOfInverseFunctionTheoremtopologicalSpaces
Date of creation 2013-03-22 13:31:55
Last modified on 2013-03-22 13:31:55
Owner paolini (1187)
Last modified by paolini (1187)
Numerical id 5
Author paolini (1187)
Entry type Proof
Classification msc 54C05