homotopy of maps

Let X,Y be topological spacesMathworldPlanetmath, A a closed subspace of X and f,g:XY continuous mapsMathworldPlanetmath. A homotopy of maps is a continuous functionMathworldPlanetmath F:X×[0,1]Y satisfying

  1. 1.

    F(x,0)=f(x) for all xX

  2. 2.

    F(x,1)=g(x) for all xX

  3. 3.

    F(x,t)=f(x)=g(x) for all xA,t[0,1].

We say that f is homotopic to g relative to A and denote this by fg relA. If A=, this can be written fg. If g is the constant map (i.e. g(x)=y for all xX), then we say that f is nullhomotopic.

Title homotopy of maps
Canonical name HomotopyOfMaps
Date of creation 2013-03-22 12:13:19
Last modified on 2013-03-22 12:13:19
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 12
Author mathcam (2727)
Entry type Definition
Classification msc 55Q05
Synonym homotopic maps
Related topic HomotopyOfPaths
Related topic HomotopyEquivalence
Related topic ConstantFunction
Related topic Contractible
Defines homotopic
Defines nullhomotopic