TITLE: Is every contractible space a cone?
QUESTION [15 upvotes]: It is easy to show that for any topological space $X$, the cone $CX$ is contractible. I am interested in the converse. If $Y$ is a contractible space, is $Y$ homeomorphic to $CX$ for some topological space $X$?
I was told the answer is no. However, I haven't been able to find a counterexample. I have two questions:

Is there a nice non-constructive way to see that counter-examples "should"/must exist?
Does anyone have a counter-example?

REPLY [9 votes]: A fun counterexample: written in 1-dimensional strokes, the letter $\pi$.