# Cone over the Join of two topological spaces

Suppose $X$ and $Y$ are topological spaces. Let's define the join $X\ast Y$ as the quotient space $X\times Y\times [0,1]/\sim$, where $\sim$ is the equivalence relation generated by $(x,y,0)\sim(x,y',0)$ and $(x,y,1)\sim(x',y,1)$. In particular, define the cone over $X$, $Cone(X)$, as the join of $X$ with a point. Is it true that $Cone(X\ast Y)$ is homeomorphic to $Cone(X)\times Cone(Y)$? If not, when does this happen?

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is there any reason to expect this to be true? Is there a non-trivial example where it is true? –  Will Sawin Mar 21 '12 at 4:09
Hint: If you forget all the coordinates from X and Y, the first space maps to a triangle and the second maps to a square; try to find a homeomorphism between those that preserves the type of preimage. –  Tyler Lawson Mar 21 '12 at 5:07
(This may be a compactly-generated hint.) –  Tyler Lawson Mar 21 '12 at 5:27
Isn't this homework ? –  BS. Mar 21 '12 at 10:09
It´s not a homework I came uo with this problem when I was traying to understand the proof of a Proposition. –  Antonio Mar 21 '12 at 13:29