Path homotopy

A path homotopy between two paths $\gamma_{0},\gamma_{1}:[0,1]\rightarrow X$ with the same endpoints is a homotopy
$$H:[0,1]\times[0,1]\rightarrow X$$such that the endpoints stay fixed:
$$H(0,s)=\gamma_{0}(0)=\gamma_{1}(0),\;\; H(1,s)=\gamma_{0}(1)=\gamma_{1}(1).$$

Remarks

• If such a path homotopy exists, we say that $\gamma_{0}$ and $\gamma_{1}$ are path-homotopic.