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Let $f:X \to W$ and $g: Y \to Z$ be quotient maps. Find an example where $f \times g : X \times Y \to W \times Z$ is not a quotient map.

I tried to let $X=W=Y=Z=\Bbb R$ and define $f: \Bbb R \to \Bbb R$ as $f(x)=x, g: \Bbb R \to \Bbb R$ as $g(x)=-x$. Now I would have the product map $$f \times g: \Bbb R^2 \to \Bbb R^2, (f \times g)(x,y)=(x,-y)$$

I think this is a surjection, but I don't know how to check if the openness condition of quotient map is satisfied or not. That is I should have $U \subset \Bbb R^2$ is open iff $(f\times g)^{-1}(U)$ is open in $\Bbb R^2$?

Emil Grönberg
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  • Your map is a homeomorphism, so it's continuous, open, closed, and a bijection. So it's a quotient map in the trivial sense (the equivalence classes that define the identifications are all singleton sets). There's no nontrivial "collapsing" happening. – Sammy Black Jul 21 '22 at 22:33
  • See comments to this answer. – Sammy Black Jul 21 '22 at 22:34

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