local homeomorphisms between real numbers


PropositionPlanetmathPlanetmath. Let I be an open interval and f:I be a continuous mapMathworldPlanetmath. Then f is a local homeomorphism if and only if f is a homeomorphism onto image.

Proof. ,,” If f is a homeomorphism onto image, then (in particular) f is monotonic and continuousMathworldPlanetmath, thus f(I) is open in (please, see this entry (http://planetmath.org/InjectiveMapBetweenRealNumbersIsAHomeomorphism) for more details). It is easy to see that therefore f is a local homeomorphism.

,,” Assume that f is not a homeomorphism onto image. It is well known, that this implies that f is not injectivePlanetmathPlanetmath (please, see this entry (http://planetmath.org/InjectiveMapBetweenRealNumbersIsAHomeomorphism) for more details). Let x,yI be such that x<y and f(x)=f(y). Then there exists cI such that x<c<y and c is a local maximumMathworldPlanetmath of f. Thus (since f is a Darboux function) for any ε>0 there are points xε,yε(c-ε,c+ε) such that f(xε)=f(yε). This obviously implies that f cannot be locally inverted around c.

Title local homeomorphisms between real numbers
Canonical name LocalHomeomorphismsBetweenRealNumbers
Date of creation 2013-03-22 18:53:50
Last modified on 2013-03-22 18:53:50
Owner joking (16130)
Last modified by joking (16130)
Numerical id 5
Author joking (16130)
Entry type Theorem
Classification msc 54C05