# homotopy of maps

Let $X,Y$ be topological spaces, $A$ a closed subspace of $X$ and $f,g:X\to Y$ continuous maps. A homotopy of maps is a continuous function $F:X\times[0,1]\to Y$ satisfying

1. 1.

$F(x,0)=f(x)$ for all $x\in X$

2. 2.

$F(x,1)=g(x)$ for all $x\in X$

3. 3.

$F(x,t)=f(x)=g(x)$ for all $x\in A,t\in[0,1]$.

We say that $f$ is homotopic to $g$ relative to $A$ and denote this by $f\simeq g$ $relA$. If $A=\emptyset$, this can be written $f\simeq g$. If $g$ is the constant map (i.e. $g(x)=y$ for all $x\in X$), 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