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 $x$.