TITLE: Is $X/A$ contractible if both $X$ and $A$ are?
QUESTION [13 upvotes]: Let $X$ be a contractible topological space. Let $A\subseteq X$ be a contractible subspace. Is the quotient space $X/A$ necessarily contractible?
It is not hard to show that this is true if, for example, the pair $(X,A)$ has the homotopy extension property (see, e.g. Proposition 0.17 of Hatcher, Algebraic Topology). 
Some friends and I amused ourselves by trying to answer this question without assuming that $(X,A)$ has the HEP, but we were unsuccessful. Any insight would be appreciated.

REPLY [5 votes]: Cannon and Conner have a counterexample in their paper "On the Fundamental Group of One-Dimensional Spaces." See Example 2.0.2. This is a doubled cone on the Hawaiian Earring space, which they show is not contractible. However, if you expand the point of contact between the two cones to a line segment, then it is contractible.