product with the interval homotopy canonically equivalent

Let \(X\) be a topological space, \(I = [0,1]\) the interval. Then

\begin{align*} X \cong X \times I \end{align*}

are homotopy equivalent

Let

\begin{align*} \iota: X \rightarrow& X \times I \\ x \mapsto& (x,0) \end{align*}

and

\begin{align*} \pi: X \times I \rightarrow& X \\ (x,j) \mapsto& x \end{align*}

Then it follows, that

\begin{align*} \pi \circ \iota =& \mathrm{id}_{X} \\ \end{align*}

and

\begin{align*} \iota \circ \pi(x,j) = (x,0) \end{align*}

Thus it remains to show, that there exists a homotopy from \(\pi \circ \iota\) to \(\mathrm{id}_{X \times I}\). Here

\begin{align*} H: [0,1] \times (X \times I) \rightarrow& (X \times I) \\ (t,(x,j)) \mapsto& (x, t \cdot j) \end{align*}

does the job.