Theorem 1

Let X be a topological space and let A be a closed subspace of X; denote by $i: A\rightarrow X$ the canonical injection. The following assertions are equivalent:

(i) The pair $(X,A)$ has the homotopy extension property ;

(ii) There exists a continuous mapping $\varphi : X\rightarrow \mathbf{I}$ such that $A =\overset{-1}{\varphi}(0)$ and a homotopy $\sigma : X\times \mathbf{I}\rightarrow X$ fixed on A whose term is the identity mapping of X and such that $\sigma (x,0)\in A$ for every point $x\in X$ such that $\varphi (x)= 1\not$ .

(iii) There exists a continuous mapping $\varphi : X\rightarrow \mathbf{R}_+$ such that $A =\overset{-1}{\varphi}(0)$ and a homotopy $\sigma :\overset{-1}{\varphi}(\mathbf{I})\times \mathbf{I}\rightarrow X$, fixed on A, such that $\sigma (x,1) =x$ and $\sigma (x,0)\in A$ for every $x\in \overset{-1}{\varphi}(\mathbf{I})$.

(iv) There exists a continuous mapping $\varphi : X\rightarrow \mathbf{R}_+$ such that $A\subset \overset{-1}{\varphi}(0)$ and a continuous mapping

such that $\sigma (x,1) =x$ for every $x\in X$ and $\sigma (x,1-\varphi (x))\in A$ for every $x\in X$ such that $\varphi (x)\leqslant 1$.

We shall denote by $i$ the canonical injection of A into X$,\alpha_i: A\times \mathbf{I}\rightarrow$ Cyl($i$) and $\beta_i: X\rightarrow$ Cyl($i$) the canonical mappings, and by $j$ the mapping of Cyl($i$) into $X\times \mathbf{I}$ such that $j(\alpha_i(x, s)) = (x, s)$ for $(x, s)\in A\times \mathbf{I}$ and $j(\beta_i(x)) = (x,1)$ for $x\in X$.

Suppose that the pair $(X,A)$ has the homotopy extension property and let $r: X\times \mathbf{I}\rightarrow$ Cyl($i$) be a continuous retraction of the mapping $j($III, p. 241, Proposition 7). Let us denote by $\sigma$ the continuous mapping pr$_1\circ j\circ r$ of $X\times \mathbf{I}$ into X. For every $(x, t)\in X\times \mathbf{I}$, one has $|$pr$_2(j(r(x, t)))-t|\leqslant 1$. Since $\mathbf{I}$ is compact, the first projection pr$_1: X\times \mathbf{I}\rightarrow X$ is a proper mapping; it is also open. By Lemma 3, the mapping $\varphi : X\rightarrow \mathbf{I}$ given by

$x\mapsto$ sup$_{t\in\mathbf{I}}(|$pr$_2(j(r(x, t)))-t|)$

is therefore continuous.

For $x\in X$, one has $\sigma (x,1) =x$. Let $x\in X$ be such that $\sigma (x,0)\notin A$; then pr$_2(j(r(x,0))) = 1$ and $\varphi (x)\geqslant 1$. For $x\in A$ and $t\in \mathbf{I}$, one has $j(r(x, t)) = (x, t)$; consequently, $\sigma (x, t) =x$ and $\varphi (x) = 0$. Conversely, let $x\in X$ be such that $\varphi (x) = 0$. One has therefore pr$_2(j(r(x, t))\leqslant t$ for every $t\in \mathbf{I}$; if $t <1$, this implies $j(r(x, t))\in A\times \mathbf{I}$; since A is closed in X, one has therefore $j(r(x,1))\in A\times \mathbf{I}$, hence $x\in A$.

This proves that (i) implies (ii).

Let $\varphi$ and $\sigma$ be mappings satisfying the properties of assertion (ii). Put $\varphi_1= 2\varphi$ and let $\sigma_1$ be the restriction of $\sigma$ to $\overset{-1}{\varphi}_{1}(\mathbf{I})\times \mathbf{I}$. We have $\overset{-1}{\varphi}_{1}(0) = A$; for $a\in A,\sigma_1(a, t) =a$ for all $a\in A$. Let $x\in X$ be such that $\varphi_1(x)\leqslant 1$; we have $\sigma_1(x,1) =x$; moreover, $\varphi (x) =\varphi_1(x)/2<1$, hence $\sigma_1(x,0)\in A$. Thus, (ii) implies (iii).

Let us prove that (iii) implies (iv). Let $\varphi : X\rightarrow \mathbf{R}$ and $\sigma :\overset{-1}{\varphi}(\mathbf{I})\times$ $\mathbf{I}\rightarrow X$ be as in the statement; put $B =\overset{-1}{\varphi}(\mathbf{I})$ and $C =\overset{-1}{\varphi}([1,+\infty [)$.

Let $u_1$ be the mapping from $B\times \mathbf{I}$ into X such that $u_1(x, t) =\sigma (x,1-$ $(1-t)/2\varphi (x))$ if $t+ 2\varphi (x)\geqslant 1$ and $\varphi (x)>0$ and $u_1(x, t) =\sigma (x,0)$ otherwise. By the following lemma 4, it is continuous.

Let $u_2$ be the mapping from $B\times \mathbf{I}$ into X such that $u_2(x, t) =$ $\sigma (x$, sup(0$,1-2(1-t)(1-\varphi (x))$)). It is continuous. For $x\in B$ such that $\varphi (x) = 1/2$ and $t\in \mathbf{I}$, we have $u_1(x, t) =\sigma (x, t) =u_2(x, t)$. Let $u: B\times \mathbf{I}\rightarrow X$ denote the mapping such that $u(x, t) =u_1(x, t)$ if $\varphi (x)\leqslant 1/2$ and $u(x, t) =u_2(x, t)$ if $1/2< x\leqslant 1$ ; it is continuous since its restrictions to the closed subspaces $\overset{-1}{\varphi}([0,1/2])\times \mathbf{I}$ and $\overset{-1}{\varphi}([1/2,1])\times \mathbf{I}$ of $B\times \mathbf{I}$ are continuous (TG, I, p. 19, prop. 4).

For $x\in X$ such that $\varphi (x) = 1$ and $t\in \mathbf{I}$, we have $u(x, t) =u_2(x, t) =$ $\sigma (x,1) =x$. There therefore exists a unique mapping $\tau : X\times \mathbf{I}\rightarrow X$ which coincides with $u$ in $B\times \mathbf{I}$ and with the mapping pr$_1$ in $C\times \mathbf{I}$; it is continuous (loc. cit.).

Let $x\in X$; it is verified that one has $\tau (x,1) = 1$. If, moreover, $x\in A$, then one has $\varphi (x) = 0$, hence $\tau (x, t) =u_1(x, t) =\sigma (x,0) =x$. Finally, if $2\varphi (x)\leqslant 1$, then $\varphi (x)\leqslant 1/2$, hence $\tau (x,1-2\varphi (x)) =\sigma (x,0)\in A$.

This proves that assertion (iv) is satisfied.

Suppose finally that assertion (iv) is satisfied, and let us prove that the pair $(X,A)$ has the homotopy extension property.

Let $C_1$ (resp. $C_2$) be the set of pairs $(x, t)\in X\times \mathbf{I}$ such that $t+\varphi (x)\leqslant 1$ (resp. $t+\varphi (x)\geqslant$ 1). These are closed sets. For $(x, t)\in C_1$, one has $\sigma (x,1-\varphi (x))\in A$; let then $\rho_1: C_1\rightarrow$ Cyl($i$) be the mapping defined by $(x, t)\mapsto \alpha_i(\sigma (x,1-\varphi (x)), t+\varphi (x))$; it is continuous. Let also $\rho_2: C_2\rightarrow$ Cyl($i$) be the continuous mapping defined by $(x, t)\mapsto \beta_i(\sigma (x, t))$. For $(x, t)\in C_1\cap C_2$, one has $t+\varphi (x) = 1$, hence

There therefore exists a unique mapping $\rho : X\times \mathbf{I}\rightarrow$ Cyl($i$) which coincides with $\rho_1$ in $C_1$ and with $\rho_2$ in $C_2$; it is continuous (TG, I, p. 19, prop. 4).

For $x\in A$ and $t\in \mathbf{I}$, one has $\varphi (x) = 0$, hence $t+\varphi (x)\leqslant 1$

For $x\in X$, one also has

Since the images of the mappings $\alpha_i$ and $\beta_i$ cover Cyl($i$), it follows that the mapping $\rho$ is a continuous retraction of the mapping $j$. Consequently, the pair $(X,A)$ has the homotopy extension property (III, p. 241, prop. 7), which concludes the proof of the theorem.