# local homeomorphism

Definition. Let $X$ and $Y$ be topological spaces. Continuous map $f:X\to Y$ is said to be locally invertible in $x\in X$ 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 $f:\mathbb{C}\to\mathbb{C}$ 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 $z\in\mathbb{C}$).

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 LocalHomeomorphism 2013-03-22 18:53:47 2013-03-22 18:53:47 joking (16130) joking (16130) 4 joking (16130) Definition msc 54C05