Definition 1

Let X be a topological space, let $c_0$ and $c_1$ be paths in X and $\sigma :\mathbf{I}\times \mathbf{I}\rightarrow X$ a homotopy joining $c_0$ to $c_1$. One says that $\sigma$ is a strict homotopy if the mappings $s\mapsto \sigma (0, s)$ and $s\mapsto \sigma (1, s)$ are constant.

One says that paths $c_0$ and $c_1$ in X are strictly homotopic if there exists a strict homotopy joining $c_0$ to $c_1$.

Two strictly homotopic paths have the same origin and the same end.