# 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 $X\to Y$ and $Z\to X$ 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 $\mathrm{id}:X\to Y$ and $\mathrm{id}:Z\to X$ are continuous, but since $X$ is neither discrete nor antidiscrete, these mappings are not homeomorphisms.

Example 2. Consider the function $f:[0,1)\to {S}^{1}$ (here ${S}^{1}$ denotes the unit circle in a complex plane^{}) defined by the formula^{} $f(t)={e}^{2\pi it}$. 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 |