LocalHomeomorphism 2009-04-18 11:40:09 2009-04-18 11:40:09 Definition homeomorphism % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \textbf{Definition.} Let $X$ and $Y$ be topological spaces. Continuous map $f:X\to Y$ is said to be \textit{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 \textit{local homeomorphism}. \textbf{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).