Exercise 27, § 1

Let $X$ be a topological space and let $A$ be a subspace (not necessarily closed) of $X$; denote by $C$ the subspace $(X\times \{0\})\cup (A\times \mathbf{I})$ of $X\times \mathbf{I}$. Suppose that $C$ is a retract of $X\times \mathbf{I}$. Let $U$ be a subset of $C$ such that $U\cap (X\times \{0\})$ and $U\cap (A\times \mathbf{I})$ are open in $X\times \{0\}$ and $A\times \mathbf{I}$ respectively. a) Let $V_0$ be the set of $x\in X$ such that $(x,0)\in U$; for $n\geqslant 1$, let $V_n$ be the union of the open subsets W of X such that $(W\cap A)\times [0,1/n[$ is contained in U.

Exercise prove that $A\cap V_0=\bigcup_{n\geqslant 1}A\cap V_n$; proving also that every open subset W of X such that $W\cap A\subset V_n$ is contained in $V_n$.

b) Exercise prove that $V\subset \bigcup_{n\geqslant 1}V_n$.

c) Exercise prove that U is open in C.

d) Exercise prove that the pair $(X,A)$ has the property of homotopy extension.