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Given a quotient map $p:X \to Y$, when is $p \times id_Z: X \times Z \to Y \times Z$ a quotient map? As shown in another question, it is sufficient for $Z$ to be locally compact. I'm now wondering about conditions on $X$ and $Y$. I know of two

  • As mentioned in the same question, the condition that $Y \cong X/A$ for $A$ a compact subspace of $X$ is also sufficient (and that the map $p$ is the induced one).
  • If $X$ and $Y$ are compact Hausdorff, then it also holds.

What are some other conditions? Ideally, I'd like for the two above conditions to be implied by only one condition. Maybe the first one implies the second one, though I'm not seeing this.

Martin Sleziak
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Fernando Chu
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1 Answers1

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A closed surjective map is a quotient map, so a sufficient (albeit not necessary) condition is that $p\times\mathrm{id}_Z$ is closed for all spaces $Z$. This property is called being "proper" (but depending on where you look, the word "proper" can also mean something different; also, this condition is sometimes called "perfect" instead). The map $p$ is proper if and only if it is a closed map and preimages of compact subsets under $p$ are compact. If $X$ is compact and $Y$ is Hausdorff, any map $p\colon X\rightarrow Y$ is proper. If $A\subseteq X$ is compact and closed (admittedly slightly weaker than the condition in your post), then $p\colon X\rightarrow X/A$ is proper.

Thorgott
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