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Let me consider a space to be locally compact, if every point has a neighborhood whose closure is compact. Consider a continuous map $f:X\to Y$ between locally compact Hausdorff spaces. Is it true that $f$ can be factored as $$X\stackrel{g}{\to}Z\stackrel{h}{\to}Y$$ where $g$ is an open embedding, i.e. $g$ is a homeomorphism onto its image, and $h$ is a proper map, i.e. preimage of compact sets under $h$ are compact. This is true for separated morphisms of quasicompact quasiseparated schemes with corresponding notions of open immersion and proper maps, which is a theorem of Deligne-Nagata. I believe this is probably also true for these spaces, but I haven't found a proof yet. Can someone provide a proof or mention a reference where this might be shown? I wouldn't be surprised if continuous is too weak to support such a property, so I would love to know if we need more conditions on $f$ to make this work?

gurgur dilon
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2 Answers2

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There's a fairly efficient—but little known—way that you can produce such a factorisation due to Whyburn called the unified space. Given $f \colon X \to Y$ let $Z = X \sqcup Y$ as sets and define $g \colon Z \to Y$ by

$$ g(z) = \begin{cases} z & \text{if } z \in Y\\ f(z) & \text{if } z \in X. \end{cases} $$

Equip $Z$ with the topology generated by open sets in $X$ and sets of the form $g^{-1}(U) \cap (Z \setminus K)$ where $U$ is open in $Y$ and $K$ is compact in $X$.

Then $Z$ is a locally compact Hausdorff space, the obvious inclusion $h \colon X \to Z$ is open, $g \colon Z \to Y$ is proper, and $f = g \circ h$.

As an added bonus, $g$ is surjective and the inclusion $Y \hookrightarrow Z$ is closed.

Zorngo
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    Very nice construction +1. It looks like its good at preserving properties such as first countability etc. – Jakobian Dec 20 '24 at 05:05
  • Although I don't know a reference other than my own notes, you can also attain a minimal solution to the problem by considering the open subset of $Y$, $$U = \bigcup{ W \mid W \text{ is a precompact open set in } Y \text{ such that } f^{-1}(\overline{W}) \text{ is compact in } X },$$ forming $Z\setminus U$, and restricting $g$ to $Z\setminus U$. – Zorngo Dec 20 '24 at 05:28
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    I would call this the cograph of $f$. – ronno Dec 20 '24 at 09:48
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    I've seen the cograph of $f$ defined to be the quotient of $X \sqcup Y$ by $f(x) \sim x$, this is slightly different. – Zorngo Dec 21 '24 at 00:14
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This is true, such maps $g, h$ and locally compact Hausdorff space $Z$ always exist. Consider the following theorem from Gillman and Jerison's Rings of continuous functions:

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Consider the extension $\tilde{f}:\beta X\to \beta \overline{f(X)}$ of $f$, according to above theorem if $Z = \tilde{f}^{-1}(\overline{f(X)})$ then $h = \tilde{f}\restriction_Z$ is a perfect map (i.e. closed continuous surjection with compact fibers; such maps are well-known to be proper) onto $\overline{f(X)}$. Note that $\overline{f(X)}$ is a closed subset of locally compact space $Y$ so locally compact, so that $\overline{f(X)}$ is open in $\beta\overline{f(X)}$, and so $Z$ is an open subset of $\beta X$, so locally compact. And since $X$ is locally compact, the embedding $g:X\to Z$ is open. Of course if $K\subseteq Y$ is compact then so is $K\cap \overline{f(X)}$, so any compact $K\subseteq Y$ will have compact preimage $h^{-1}(K)$.

Jakobian
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