Let $(X,A)$ be a $CW$ pair(assume X is connected) such that the embedding $i: A \hookrightarrow X$ is a homotopic equivalence, do we always have the quotient space $X/A$ is contractible? (Hatcher uses something like this in his book, but I don't know if this general proposition holds)
Moreover, if this is true, is the condition that $(X,A)$ is a CW pair necessary here? Can we simply assume $(X,A)$ is a pair of space and its subspace?
Just quoting from Hatcher here: If $(X,A)$ is a CW pair, then $(X,A)$ has the homotopy extension property (0.16). Hence, if the inclusion $i : A \to X$ is a homotopy equivalence, then $A$ is a deformation retract of $X$ (0.20).
But once you know that $A$ is a deformation retract of $X$, the result is easy. Let $p: X \to A$ be the retraction, obeying the condition that $p|_A = {\rm id}_A$, and let $F:X\times[0,1]\to X$ be the homotopy between $i \circ p $ and ${\rm id}_X$ obeying $F|_{A \times [0,1]} = {\rm id}_{A\times [0,1]}$. Then $i$ and $p$ descend to well-defined continuous maps $\tilde i: A/A \to X/A$ and $\tilde p : X/A \to A/A$, and the homotopy $F$ descends to a homotopy $\tilde F$ between $\tilde i \circ \tilde p$ and ${\rm id}_{X/A}$. So $X/A$ and $A/A$ are homotopy equivalent. But $A/A$ is a point, so $X/A$ is contractible.