# Quotient map between homotopy equivalent spaces

$$\require{AMScd}$$ Let $$X$$ be some top. space and $$X/{\raise.17ex\hbox{\scriptstyle\sim}}$$ a quotient space with quotient map $$q$$: $$\begin{CD} X @>q>> X/{\raise.17ex\hbox{\scriptstyle\sim}} \end{CD}$$ such that X is homotopy equivalent (or homeomorphic if that makes a difference) to $$X/{\raise.17ex\hbox{\scriptstyle\sim}}$$. Is it true that $$q$$ is then a homotopy equivalence?

Thinking about 'nice' examples this seemed reasonable, at least I didn't see an example where it goes wrong. Trying to prove it though, there didn't seem to be any reasoning why $$q$$ should be homotopic to the given homotopy equivalence. So I assume this doesn't hold in general? Is it perhaps true for 'nice' spaces such as CW-complexes?

Consider, for example, the discrete countable space $$\mathbb{N}$$. Given any partition of $$\mathbb{N}$$ into finite subsets $$\mathbb{N}=\coprod_{n \in \mathbb{N}}F_n$$, there is the quotient $$X/\sim$$ where $$\sim$$ is the equivalence such that $$F_n$$'s are its equivalence classes, i.e. $$x \sim y$$ iff $$x, y \in F_n$$for some $$n$$. Then $$X$$ and $$X/\sim$$ are homeomorphic (for example by $$n \mapsto F_n,$$ both are countable discrete spaces). But the quotient map is not a homotopy equivalence, because it does not induce bijection on the set of path-components, which are singletons in both cases, unless all the sets $$F_n$$ are themselves singletons.
Similarly, you can take e.g. $$X=\mathbb{C}\setminus \mathbb{Z}$$, and the quotient $$X/\sim$$ obtained by shrinking $$\{z \in \mathbb{C}\;|\; 0\neq |z| \leq\frac{1}{2}\}$$ into a point. Then again, $$X/\sim$$ is homeomorphic to $$X$$ (both are a plane with countable set of isolated punctures), but the quotient map does not induce isomorphism on fundamental groups, because the non-trivial loop that goes once around the origin in X becomes trivial after applying $$q$$.
If $$(X,A)$$ is a $$CW$$ pair consisting of a $$CW$$ complex $$X$$ and a contratible subcomplex $$A,$$ then the quotient map $$X\to X/A$$ is a homotopy equivalence.