Is $X/A$ contractible if both $X$ and $A$ are?

general-topology

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.

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