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?
Best Answer
I think there's something wrong with your statement that $(X,A)$ has the homotopy extension property if $X$ is Hausdorff and $A$ is closed. Take the Hawaiian earring space $E$ and form a new space $X$ by joining two copies of $E$ by an edge from the wedge point to the wedge point. Then contracting the central edge to get a new space $E\vee E$ is not a homotopy equivalence. For example, it is not surjective on the fundamental group. There are loops in $E\vee E$ which travel back and forth over each copy of $E$ infinitely many times. However there are no such paths in $X$ since they would have to travel over the central edge infinitely many times.
Hatcher assumes $(X,A)$ are a CW pair, which is a far stronger condition.