$X \times Y$ homeomorphic to $Z \times Y$ implies $X$ is homeomorphic to $Z$?

Question

True or false, if true, prove if false provide counterexample.

If $X \times Y$ is homeomorphic to $Z \times Y$ (in the product topology) then $X$ is homeomorphic to $Z$.

Need help. I've been stuck on this for awhile. I think it's true but I can't prove it...

Standard counterexample: $X= \Bbb{R}, Y= \Bbb{R}^{\omega} , Z= { 0 }$. — Crostul, Jun 09 '16 at 14:57

score 9 · Accepted Answer · answered Jun 09 '16 at 14:53

9

Let $X = \{0\}$, $Y = Z = \mathbb{N}$ each with discrete topology. Then we only have to see that there is a bijection between $X \times Y$ and $Z \times Y$ but there is no bijection between $X$ and $Z$. But this should be known.

answered Jun 09 '16 at 14:53

So we use the discrete topology to ensure that any bijection is a homeomorphism? – Tuo Jun 09 '16 at 15:49
Yes, that was my idea. – Jun 09 '16 at 16:03

score 3 · Answer 2 · answered Jun 09 '16 at 17:17

3

You can generate a whole family of examples in the following way. Let $X$ be any space that is not homeomorphic to $X^{\Bbb N}$; familiar nice examples include $\{0,1\}$ with the discrete topology, $[0,1]$, $\Bbb Z$, and $\Bbb R$. Let $Y=Z=X^{\Bbb N}$. Then

$$X\times Y\cong X^{\Bbb N}\cong X^{\Bbb N\times\Bbb N}\cong Z\times Y\;,$$

but $X\not\cong Z$.

answered Jun 09 '16 at 17:17

Brian M. Scott

631,399

I know this is an old post but what if $Y=[0,1]$? Just wondering if there are any simple examples for $X$ and $Z$. – user85798 Jan 29 '18 at 17:15

$X \times Y$ homeomorphic to $Z \times Y$ implies $X$ is homeomorphic to $Z$?

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