homotopy of paths
Let $X$ be a topological space^{} and $p,q$ paths in $X$ with the same initial point ${x}_{0}$ and terminal point ${x}_{1}$. If there exists a continuous function^{} $F:I\times I\to X$ such that

1.
$F(s,0)=p(s)$ for all $s\in I$

2.
$F(s,1)=q(s)$ for all $s\in I$

3.
$F(0,t)={x}_{0}$ for all $t\in I$

4.
$F(1,t)={x}_{1}$ for all $t\in I$
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  20130322 12:13:16 
Last modified on  20130322 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 