TITLE: Is the quotient map a homotopy equivalence?
QUESTION [13 upvotes]: It is well known that, if $A \subset X$ is a reasonable contractible subspace, then the quotient map $X \to X/A$ is a homotopy equivalence ("reasonable" means that the pair $(X,A)$ has the homotopy extension property, e.g. if it is a CW pair). For example, it's proposition 0.17 in Hatcher's celebrated Algebraic Topology.
It seems to me that this result should be a consequence of the following result, which seems true but for which I've been unable to find a proof or a reference.
Proposition? If $X$ is Hausdorff and $\sim$ is an equivalence relation whose classes $C$ are contractible and such that every pair $(X, C)$ has the homotopy extension property, then the quotient map $X \to X/\sim$ is a homotopy equivalence.
The classical result would of course be a direct corollary of this one. 
So, is this proposition true?

REPLY [3 votes]: The conjectured proposition is false. Even though it is not the main subject of the paper, a counter-example is given by the "Example 2.11" in the paper On fundamental groups of quotient spaces by Calcut, Gompf and McCarthy. They give a more detailed description of this example in section 3.7 of their paper Quotient maps with connected fibers and the fundamental group. For the sake of completeness, I give the highlights of this counter-example below.
Let $H = \{(x,y,z) \in \mathbb{R}^3 \, | \, z \ge 0\}$. It is an Hausdorff contractible space. It admits an equivalence relation $\sim$ such that all classes are properly smoothly embedded copies of $[0,1]$ or of $\mathbb{R}$ which are closed in $H$; As such, given any class $C$, $C$ is contractible and the pair $(H, C)$ is a good pair, so it has the homotopy extension property. However, the quotient $H/\sim$ is homeomorphic to $S^2$ which is not contractible, hence not homotopy equivalent to $H$.
In fact, if $\sigma : S^3 \setminus \{(0,0,0,1)\} \to \mathbb{R}^3$ denotes the stereographic projection, then $\sim$ is the image by $\sigma$ of the relation induced on $\sigma^{-1}(H)$ by the Hopf fibration.