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 $f:ℂ\to ℂ$ 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 $z\in ℂ$).

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).