homotopy of pathsHomotopyOfPaths2002-01-23 13:39:042002-02-12 08:05:30Definition% this is the default PlanetMath preamble. as your knowledge
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% define commands hereLet $X$ 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
\begin{enumerate}
\item $F(s,0)=p(s)$ for all $s \in I$
\item $F(s,1)=q(s)$ for all $s \in I$
\item $F(0,t)=x_{0}$ for all $t \in I$
\item $F(1,t)=x_{1}$ for all $t \in I$
\end{enumerate}
we call $F$ a \emph{homotopy of paths} in $X$ and say $p,q$ are \emph{homotopic paths} in $X$. $F$ is also called a \emph{continuous deformation}.