Take the 2-minute tour ×  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?
share|improve this question So the cylinder is $X \times I$
while the cone is $(X \times I )/ (X \times \{0\})$. –  john
mangual Oct 9 '12 at 15:10 add comment  1 Answer  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.  share|improve this a