When is cone homeomorphic to cylinder

Question

This is the problem: Find non-trivial example of (linaer connected) space X so that cone over X is homeomorphic to cylinder over X. Trivial examples are one point set and empty set. I have absolutely no idea.My friend says it's an interval, then cone is triangle and cylinder is a rectangle, but I don't agree that they are homeomorphic. Am I wrong?

Answer 1

For a closed interval, the cone and cylinders are a closed triangle and rectangle, and those are homeomorphic because either is homeomorphic to a closed disk.

For an open interval, however, you get into trouble. Then then the cone contains an isolated boundary point but the cylinder doesn't.

For a half-open interval there's still trouble, because the boundary of the cone is a half-open interval but the boundary of the cylinder is a closed interval.