# local homeomorphism

Definition. Let $X$ and $Y$ be topological spaces^{}. Continuous map^{} $f:X\to Y$ is said to be locally invertible in $x\mathrm{\in}X$ iff there exist open subsets $U\subseteq X$ and $V\subseteq Y$ such that $x\in U$, $f(x)\in V$ and the restriction^{}

$$f:U\to V$$ |

is a homeomorphism. If $f$ is locally invertible in every point of $X$, then $f$ is called a local homeomorphism.

Examples. Of course every homeomorphism is a local homeomorphism, but the converse^{} is not true. For example, let $f:\u2102\to \u2102$ be an exponential function^{}, i.e. $f(z)={e}^{z}$. Then $f$ is a local homeomorphism, but it is not a homeorphism (indeed, $f(z)=f(z+2\pi i)$ for any $z\in \u2102$).

One of the most important theorem of differential calculus (i.e. inverse function theorem^{}) states, that if $f:M\to N$ is a ${C}^{1}$-map between ${C}^{1}$-manifolds such that ${T}_{x}f:{T}_{x}M\to {T}_{f(x)}N$ is a linear isomorphism for a given $x\in M$, then $f$ is locally invertible in $x$ (in this case the local inverse^{} is even a ${C}^{1}$-map).

Title | local homeomorphism |
---|---|

Canonical name | LocalHomeomorphism |

Date of creation | 2013-03-22 18:53:47 |

Last modified on | 2013-03-22 18:53:47 |

Owner | joking (16130) |

Last modified by | joking (16130) |

Numerical id | 4 |

Author | joking (16130) |

Entry type | Definition |

Classification | msc 54C05 |