example of continuous bijections which are not homeomorphisms
Example 1. Assume that X is a topological space, which neither discrete nor antidiscrete. We will show that there are topological spaces Y and Z such that there are continuous
bijections X→Y and Z→X 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:X→Y and id:Z→X 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 plane) defined by the formula
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 compact
).
Title | example of continuous bijections which are not homeomorphisms |
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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 |