loop space (original) (raw)

Let X be a topological space, and give the space of continuous maps [0,1]→X, the compact-open topology, that is a subbasis for the topology is the collection of sets {σ:σ⁢(K)⊂U} for K⊂[0,1] compact and U⊂X open.

Then for x∈X, let Ωx⁢X be the subset of loops based at x (that is σ such that σ⁢(0)=σ⁢(1)=x), with the relative topology.