loop space LoopSpace 2002-02-02 03:31:14 2003-07-25 10:45:35 Definition % 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 Let $X$ be a topological space, and give the space of continuous maps $[0,1]\to X$, the compact-open topology, that is a subbasis for the topology is the collection of sets $\{\sigma : \sigma(K)\subset U \}$ for $K\subset [0,1]$ compact and $U\subset X$ open. Then for $x\in X$, let $\Omega_{x}X$ be the subset of loops based at $x$ (that is $\sigma$ such that $\sigma(0) = \sigma(1) = x$), with the relative topology. $\Omega_{x}X$ is called the loop space of $X$ at $x$.