example of continuous bijections which are not homeomorphisms

Example 1. Assume that X is a topological spaceMathworldPlanetmath, which neither discrete nor antidiscrete. We will show that there are topological spaces Y and Z such that there are continuousPlanetmathPlanetmath bijections XY and ZX which are not homeomorphisms.

Let Y=Z=X as a sets but topology on Y is antidiscrete and on Z is discrete. Then obviously identity mappings id:XY and id:ZX are continuous, but since X is neither discrete nor antidiscrete, these mappings are not homeomorphisms.

Example 2. Consider the function f:[0,1)S1 (here S1 denotes the unit circle in a complex planeMathworldPlanetmath) defined by the formulaMathworldPlanetmathPlanetmath f(t)=e2πit. It is easy to see that f is a continuous bijection, but f is not a homeomorphism (because [0,1) is not compactPlanetmathPlanetmath).

Title example of continuous bijections which are not homeomorphisms
Canonical name ExampleOfContinuousBijectionsWhichAreNotHomeomorphisms
Date of creation 2013-03-22 18:54:31
Last modified on 2013-03-22 18:54:31
Owner joking (16130)
Last modified by joking (16130)
Numerical id 4
Author joking (16130)
Entry type Example
Classification msc 54C05