loop space

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

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

ΩxX is called the loop spaceMathworldPlanetmath 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 SuspensionMathworldPlanetmath
Related topic EilenbergMacLaneSpace