(a) What does it mean for two spaces $X$ and $Y$ to be homotopy equivalent?

(b) What does it mean for a subspace $Y \subseteq X$ to be a retract of a space $X$ ? What does it mean for a space $X$ to be contractible? Show that a retract of a contractible space is contractible.

(c) Let $X$ be a space and $A \subseteq X$ a subspace. We say the pair $(X, A)$ has the homotopy extension property if, for any pair of maps $f: X \times\{0\} \rightarrow Y$ and $H^{\prime}: A \times I \rightarrow Y$ with

$$\left.f\right|_{A \times\{0\}}=\left.H^{\prime}\right|_{A \times\{0\}},$$

there exists a map $H: X \times I \rightarrow Y$ with

$$\left.H\right|_{X \times\{0\}}=f,\left.\quad H\right|_{A \times I}=H^{\prime}$$

Now suppose that $A \subseteq X$ is contractible. Denote by $X / A$ the quotient of $X$ by the equivalence relation $x \sim x^{\prime}$ if and only if $x=x^{\prime}$ or $x, x^{\prime} \in A$. Show that, if $(X, A)$ satisfies the homotopy extension property, then $X$ and $X / A$ are homotopy equivalent.