example of continuous bijections which are not homeomorphisms
Example 1. Assume that is a topological space, which neither discrete nor antidiscrete. We will show that there are topological spaces and such that there are continuous bijections and which are not homeomorphisms.
Let as a sets but topology on is antidiscrete and on is discrete. Then obviously identity mappings and are continuous, but since is neither discrete nor antidiscrete, these mappings are not homeomorphisms.
Example 2. Consider the function (here denotes the unit circle in a complex plane) defined by the formula . It is easy to see that is a continuous bijection, but is not a homeomorphism (because is not compact).
|Title||example of continuous bijections which are not homeomorphisms|
|Date of creation||2013-03-22 18:54:31|
|Last modified on||2013-03-22 18:54:31|
|Last modified by||joking (16130)|