It is known that if pair $(X,A)$ has the homotopy extension property and $A$ is contractible, then the quotient map $q:X\to X/A$ is a homotopy equivalence.
I am wondering what if the homotopy extension property is omitted. What would be a counterexample? Namely, I am looking for a space $X$ and a contractible subspace $A$ such that the quotient map $q:X\to X/A$ is not a homotopy equivalence. I am thinking about the Hawaiian Earring but am not sure if it is a correct counterexample.