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?