# 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. 1.

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

2. 2.

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

3. 3.

$F(0,t)=x_{0}$ for all $t\in I$

4. 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 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