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The following is Problem 2.2.14 in Tammo tom Dieck's Algebraic Topology:

Let $Y,Z$ be compact or $X,Z$ be locally compact. Then the canonical bijection $(X\wedge Y)\wedge Z\to X\wedge(Y\wedge Z)$ is a homeomorphism.

I'm not working in the category of compactly generated spaces. I want to use this exercise to show that the iterated suspension is $\Sigma^nX\cong X\wedge S^n$ for any space $X$. So it remains to prove this exercise.

I could prove this for the locally compact case (using the fact that the product of a quotient map with the identity map of a locally compact space is again a quotient map), but for the case $Y,Z$ compact I have no idea how to do this. Any hints will be appreciated!

Yuxiao Xie
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  • I'm so confused. What do you want to show? The implication that 2.2.14 implies that $\Sigma^n X \cong X \wedge S^n$ for any space $X$? Or do you want to prove 2.2.14? – mathworker21 Dec 21 '18 at 00:21
  • @mathworker21 Sorry for the confusion, I've edited the question. I can't prove 2.2.14... I mention $\Sigma^n X \cong X \wedge S^n$ to explain that this restricted form of associativity really has some use. That's why I want to prove the associativity. – Yuxiao Xie Dec 21 '18 at 00:26

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