injective map between real numbers is a homeomorphism


Lemma. Assume that I is an open interval and f:I is an injectivePlanetmathPlanetmath, continuous mapMathworldPlanetmath. Then f(I) is an open subset.

Proof. Since f is injective, then of course f is monotonicPlanetmathPlanetmath. Without loss of generality, we may assume that f is increasing. Let y=f(x)f(I). Since I is open, then there are α,βI such that α<x<β. Therefore f(α)<y<f(β) and (because continuous functionsMathworldPlanetmath are Darboux functions) for any y(f(α),f(β)) there exists xI such that f(x)=y. This shows that (f(α),f(β)) is an open neighbourhood of y contained in f(I) and therefore (since y was arbitrary) f(I) is open.

PropositionPlanetmathPlanetmath. Assume that I is an open interval and f:I is an injective, continuous map. Then f is a homeomorphism onto image.

Proof. Of course, it is enough to show that f is an open map. But if UI is open, then there are disjoint, open intervals Iα such that

U=αIα.

Therefore we obtain continuous, injective maps fα:Iα which are restrictionsPlanetmathPlanetmathPlanetmath of f to Iα. By lemma we have that fα(Iα) is open and therefore

f(U)=f(αIα)=αf(Iα)=αfα(Iα)

is open. This shows that f is a homeomorphism onto image.

Title injective map between real numbers is a homeomorphism
Canonical name InjectiveMapBetweenRealNumbersIsAHomeomorphism
Date of creation 2013-03-22 18:53:58
Last modified on 2013-03-22 18:53:58
Owner joking (16130)
Last modified by joking (16130)
Numerical id 4
Author joking (16130)
Entry type Theorem
Classification msc 54C05