why $(X\cup CA)/CA=X/A$ and $(X\cup CA)/X=\Sigma A$ where $CA$ is the cone on $A$ and $\Sigma A$ is the suspension of $A$.
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In the first one, there's an obvious bijection, and this bijection is easily seen to be a homeomorphism. For the second one, I think you have a typo: it should be $(X\cup CA)/X \cong \Sigma A$. This has an obvious bijection, too.