homotopy of paths

Let X be a topological spaceMathworldPlanetmath and p,q paths in X with the same initial point x0 and terminal point x1. If there exists a continuous functionMathworldPlanetmath F:I×IX such that

  1. 1.

    F(s,0)=p(s) for all sI

  2. 2.

    F(s,1)=q(s) for all sI

  3. 3.

    F(0,t)=x0 for all tI

  4. 4.

    F(1,t)=x1 for all tI

we call F a homotopy of paths in X and say p,q are homotopic paths in X. F is also called a continuous deformation.

Title homotopy of paths
Canonical name HomotopyOfPaths
Date of creation 2013-03-22 12:13:16
Last modified on 2013-03-22 12:13:16
Owner RevBobo (4)
Last modified by RevBobo (4)
Numerical id 8
Author RevBobo (4)
Entry type Definition
Classification msc 55Q05
Synonym homotopic paths
Synonym continuous deformation
Synonym homotopy
Related topic HomotopyOfMaps
Related topic HomotopyWithAContractibleDomain
Related topic PathConnectnessAsAHomotopyInvariant