# 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 $\Omega_{x}X$ 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 $x$.

Title loop space LoopSpace 2013-03-22 12:15:26 2013-03-22 12:15:26 mathcam (2727) mathcam (2727) 8 mathcam (2727) Definition msc 54-00 Suspension EilenbergMacLaneSpace