loop space

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 ${\mathrm{\Omega }}_{x}X$ be the subset of loops based at $x$ (that is $\sigma$ such that $\sigma (0)=\sigma (1)=x$), with the relative topology.

${\mathrm{\Omega }}_{x}X$ is called the loop space of $X$ at $x$.

Title	loop space
Canonical name	LoopSpace
Date of creation	2013-03-22 12:15:26
Last modified on	2013-03-22 12:15:26
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	8
Author	mathcam (2727)
Entry type	Definition
Classification	msc 54-00
Related topic	Suspension
Related topic	EilenbergMacLaneSpace