Lemma 4

Let X and Y be topological spaces, let $\varphi : X\rightarrow \mathbf{R}_+$ be a continuous mapping, and put $A =\overset{-1}{\varphi}(0)$. Let $\sigma : X\times \mathbf{I}\rightarrow Y$ be a homotopy which is fixed on A. The mapping $\sigma ': X\times \mathbf{I}\rightarrow Y$ which maps $(x, s)$ to $\sigma (x, s/\varphi (x))$ if $s < \varphi (x)$ and to $\sigma (x,1)$ if $s\geqslant \varphi (x)$ is continuous.

The mapping $\sigma '$ is continuous at every point of the closed subset $\overset{-1}{\varphi}([1,+\infty [)\times \mathbf{I}$; it is therefore enough to prove that its restriction to $\overset{-1}{\varphi}(\mathbf{I})\times \mathbf{I}$ is continuous. We may therefore suppose that $\varphi (X)\subset \mathbf{I}$. Let C and $C'$ be the subspaces of $X\times \mathbf{I}$ consisting of pairs $(x, s)$ such that $s\leqslant \varphi (x)$ and $s\geqslant \varphi (x)$ respectively. They are closed and cover $X\times \mathbf{I}$. The mapping $\sigma '$ is continuous on $C'$; let us prove that it is continuous on C.

Let $\alpha : X\times \mathbf{I}\rightarrow X\times \mathbf{I}$ be the continuous mapping defined by $\alpha (x, t) =$ $(x, t\varphi (x))$. Its image is equal to C and $\sigma '\circ \alpha$ is the continuous mapping $\sigma$. Let us prove that $\alpha$ is a proper map. Consider in fact an ultrafilter $\mathfrak{U}$ on $X\times \mathbf{I}$ and a point $(x, t)\in X\times \mathbf{I}$ adherent to the ultrafilter basis $\alpha (\mathfrak{U})$. Since pr$_1\circ \alpha =$ pr$_1$, the ultrafilter basis pr$_1(\mathfrak{U})$ on X converges to $x$. Since $\mathbf{I}$ is compact, there exists a point $s\in \mathbf{I}$ such that the ultrafilter basis pr$_2(\mathfrak{U})$ converges to $s$. Then $\mathfrak{U}$ converges to $(x, s)$. Since $\alpha$ is continuous, the ultrafilter basis $\alpha (\mathfrak{U})$ converges to $(x, s\varphi (x))$. Since $\mathbf{I}$ is separated, we have $s\varphi (x) =t$, whence $\alpha (x, s) = (x, t)$, so that $\alpha$ is proper (TG, I, p. 75, Theorem 1). It then follows from I, p. 18, example 2 and Proposition 9 that $\sigma '|C$ is continuous.