Corollary 2

Let X and Y be topological spaces, let A be a closed subspace of X, let B be a closed subspace of Y. If the pairs $(X,A)$ and $(Y,B)$ possess the homotopy extension property, the same is true of the pair $(X\times Y,(X\times B)\cup (A\times Y))$.

Let $\varphi : X\rightarrow \mathbf{R}_+$ and $\sigma :\overset{-1}{\varphi}(\mathbf{I})\times \mathbf{I}\rightarrow$ X, resp. $\varphi ': Y\rightarrow \mathbf{R}_+$

and $\sigma ':\overset{-1}{\psi}(\mathbf{I})\times \mathbf{I}\rightarrow Y$, satisfy the conditions of assertion (iv) of theorem 1 for the pair $(X,A)$, resp. for the pair $(Y,B)$. Let $\psi : X\times Y\rightarrow \mathbf{R}_+$ be the mapping given by $(x, y)\mapsto$ inf($\varphi (x), \varphi '(x)$) ; it is continuous; one has also $\psi (x, y) = 0$ for every $(x, y)\in X\times Y$ such that $x\in A$ or $y\in B$. For $(x, y, t)\in X\times Y\times \mathbf{I}$ such that $t+\psi (x)\geqslant 1$, one has $t+\varphi (x)\geqslant 1$ and $t+\varphi '(x)\geqslant 1$. Thus one defines a continuous mapping

by putting $\tau (x, y, t) = (\sigma (x, t), \sigma '(y, t))$. For every $(x, y)\in X\times Y$, one has $\tau (x, y,1) = (\sigma (x,1), \sigma '(y,1)) = (x, y)$. If, moreover, $\psi (x, y)\leqslant 1$, then $\varphi (x)\leqslant 1$ or $\varphi '(y)\leqslant 1$, so that $\sigma (x,1)\in A$ or $\sigma '(y,1)\in B$; it follows that $\tau (x, y,1-\psi (x))$ belong to $(A\times Y)\cup (X\times B)$. This verifies assertion (iv) of theorem 1, whence the corollary.