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$\require{AMScd}$ 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?

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There should be enough counterexamples to this when your space is "big enough" (but otherwise well-behaved).

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 $q$.

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Here is (only) a partial answer regarding CW complexes. It's proposition 0.17 in Hatcher's book:

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.

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