Corollary 5

Let $X'$ be a normal topological space, let X be a subspace of $X'$, let A be a closed subspace of $X'$ contained in X, and let U be a neighbourhood of A in X.

Let Y and Z be topological spaces; let $f: X\times \{1\} \times Y\rightarrow$ $X\times \{1\} \times Z$ be an $X\times \{1\}$-morphism and let $g: U\times \mathbf{I}\times Y\rightarrow U\times \mathbf{I}\times Z$ be a $U\times \mathbf{I}$-morphism which coincides with $f$ on $U\times \{1\} \times Y$.

There then exists an $X\times \mathbf{I}$-morphism $h: X\times \mathbf{I}\times Y\rightarrow X\times \mathbf{I}\times Z$ which coincides with $f$ on $X\times \{1\} \times Y$ and with $g$ on $A\times \mathbf{I}\times Y$.

Moreover, if $f$ and $g$ are homeomorphisms, one can choose a homeomorphism $h$ having the required properties.

Let $U'$ be an open neighbourhood of A in $X'$ such that $U'\cap X\subset U$. Since the space $X'$ is normal, there exists an open neighbourhood $V'$ of A in $X'$ such that $A\subset V'\subset \overline{V'}\subset U'$ (TG, IX, p. 41). Replacing U by $\overline{V'}\cap X$ if necessary, we may thus suppose that U is closed in X. Let us then take up again the notation of Prop. 1 and of its proof; let $r: X\times \mathbf{I}\rightarrow$ Cyl($i_U$) be a continuous mapping such that $j_U\circ r(x) =x$ for every point $x\in j_A$(Cyl($i_A$)).

Put also $f'=$ pr$_3\circ f$ and $g'=$ pr$_3\circ g$. Since $f'$ and $g'$ coincide on $U\times \{1\} \times Y$, there exists a unique mapping

$\varphi :$ Cyl($i_U$)$\times Y\rightarrow$ Cyl($i_U$)$\times Z$

such that $\varphi (\alpha (x, t), y) = (\alpha (x, t), g'(x, t, y))$ for $(x, t, y)\in U\times \mathbf{I}\times Y$ and $\varphi (\beta (x), y) = (\beta (x), f'(x,1, y))$ for $(x, y)\in X\times Y$. Since U is closed in X, the canonical surjection $\pi$ of $(U\times \mathbf{I})\cup X$ onto Cyl($i_U$) is universally strict (III, p. 239, remark 2). Since the mapping $\varphi \circ (\pi \times$ Id$_Y)$ is continuous, the mapping $\varphi$ is continuous. It is therefore a morphism of Cyl($i_U$)-spaces, and even an isomorphism if $f$ and $g$ are homeomorphisms.

Let then $h: X\times \mathbf{I}\times Y\rightarrow X\times \mathbf{I}\times Z$ be the mapping $r^*(\varphi )$ deduced from $\varphi$ by the base change $r$. It is given by $h(x, t, y) =$ $(x, t$, pr$_2(\varphi (r(x, t), y)))$ for $(x, t, y)\in X\times \mathbf{I}\times Y$. It is a morphism of $X\times \mathbf{I}$-spaces, and even an isomorphism if $\varphi$ is one. For $(x, t, y)\in$ $A\times \mathbf{I}\times Y$, one has $r(x, t) = (x, t)$, whence

$h(x, t, y) = (x, t$, pr$_2(\varphi (x, t, y))) = (x, t, g'(x, t, y)) =g(x, t, y)$,

which proves that $h$ coincides with $g$ on $A\times \mathbf{I}\times Y$. Analogously, for $x\in X$ and $y\in Y$, we have $r(x,1) = (x,1)$, hence

$h(x,1, y) = (x,1$, pr$_2(\varphi (x,1, y))) = (x,1, f'(x,1, y)) =f(x,1, y)$

so that $h$ coincides with $f$ on $X\times \{1\} \times Y$. The corollary is thus proved.