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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?

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So the cylinder is $X \times I$ while the cone is $(X \times I )/ (X \times \{0\})$. – cactus314 Oct 9 '12 at 15:10

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.

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