Exercise 9, § 1

If X and Y are topological spaces and $\mathscr{U}$ a set of subsets of X, one says that mappings $f, g: Y\rightarrow X$ are $\mathscr{U}$ -close if for every point $y\in Y$, there exists $U\in \mathscr{U}$ such that $f(y)$ and $g(y)$ belong to U; one says that a homotopy $\sigma : Y\times \mathbf{I}\rightarrow X$ is $\mathscr{U}$ -small if for every point $y\in Y$, there exists $U\in \mathscr{U}$ such that $\sigma (y, t)\in U$ for every $t\in \mathbf{I}$.

Let X be a closed subspace of a convex subset Z of $\mathbf{I}^{\mathbf{N}}($cf. exercise 8) ; suppose that X is an absolute neighbourhood retract.

a) Let $\mathscr{U}$ be a covering of X by open subsets of $\mathbf{I}^{\mathbf{N}}$. Prove that there exists a family $\mathscr{V}$ of open and convex subsets of $\mathbf{I}^{\mathbf{N}}$ such that, for every open set $v\in \mathscr{V}$, there exists an open set $u\in \mathscr{U}$ such that $v\cap X\subset u$, and a continuous retraction $r$ of the canonical injection of X into the union V of the family $\mathscr{V}$. b) Let Y be a topological space and let B be a closed subspace of Y. Let $f, g: Y\rightarrow X$ be continuous mappings which are $\mathscr{V}$ -close, let $\sigma : B\times \mathbf{I}\rightarrow X$ be a $\mathscr{V}$ -small homotopy whose origin is equal to $f|_B$.

Prove that there exists a neighbourhood W of B in Y and a continuous mapping $h$ of the subspace $(Y\times \{0,1\})\cup (W\times \mathbf{I})$ of $Y\times \mathbf{I}$. into X such that $h(y,0) =f(y)$ and $h(y,1) =g(y)$ for every $y\in Y$, and $h(y, t) =\sigma (y, t)$ for every $y\in B$ and every $t\in \mathbf{I}$. Prove that there exists a neighbourhood $W'$ of B contained in W such that $h|W'\times \mathbf{I}$ is a $\mathscr{V}$ -small homotopy.

c) Let $u: Y\rightarrow \mathbf{I}$ be a continuous mapping such that $u(y) = 1$ for $y\in B$ and $u(y) = 0$ in a neighbourhood of Y-W ; put

$\{(1-u(y))((1-t)f(y) +tg(y)) +u(y)h(y, t)$ if $y\in V$

$(1-t)f(y) +tg(y)$ otherwise.

Prove that $h'$ is continuous and that its image is contained in V.

d) Deduce that there exists a homotopy $\widetilde{\sigma}: Y\times \mathbf{I}\rightarrow X$ with origin $f$, with term $g$, and which is $\mathscr{U}$ -small.