Exercise 10, § 1

Let $X$ be a topological space. Suppose that there exists an open covering $\mathscr{U}$ of X such that for every topological space Y and every closed subspace B of Y, two mappings $f_0, f_1$ from Y into X which are $\mathscr{U}$ -close are homotopic if there exists a $\mathscr{U}$ -small homotopy joining $f_0|B$ to $f_1|B$.

a) Let $a$ be a point of X and let U be an element of $\mathscr{U}$ such that $a\in U$. Prove that there exists a homotopy $\sigma : U\times \mathbf{I}\rightarrow X$ such that $\sigma (x,0) =a,\sigma (x,1) =x$ and $\sigma (a, t) =a$ for all $x\in X$ and all $t\in \mathbf{I}$.

b) Let V be an open neighbourhood of $a$ such that $\sigma (V\times \mathbf{I})\subset U$. Prove that V is an absolute neighbourhood retract.

c) Prove that X is an absolute neighbourhood retract.