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

2.
$F(x,1)=g(x)$ for all $x\in X$

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 $A=\mathrm{\varnothing}$, 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  20130322 12:13:19 
Last modified on  20130322 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 