homeomorphisms preserve connected components

Let X,Y be topological spacesMathworldPlanetmath and X=Xi, Y=Yj be decompositions into connected componentsMathworldPlanetmathPlanetmathPlanetmathPlanetmath.

PropositionPlanetmathPlanetmath. Assume that f:XY is a homeomorphismPlanetmathPlanetmath. Then for any i there exists j such that f(Xi)=Yj.

Proof. Take any i. Because f is continuousPlanetmathPlanetmath f(Xi) is connected, then there exists j such that f(Xi)Yj (because Yj is a connected component). Now f is a homeomorphism, f-1(Yj)Xi, Yj is connected and Xi is a connected component, so f-1(Yj)Xi. Thus Yjf(Xi), which completesPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath the proof.

Title homeomorphisms preserve connected components
Canonical name HomeomorphismsPreserveConnectedComponents
Date of creation 2013-03-22 18:45:32
Last modified on 2013-03-22 18:45:32
Owner joking (16130)
Last modified by joking (16130)
Numerical id 5
Author joking (16130)
Entry type DerivationMathworldPlanetmath
Classification msc 54D05