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Is it true that, in the category $\mathbf{Top}$ of topological spaces and continuous maps, the compactly generated weakly Hausdorff spaces are precisely the spaces arising as filtered colimits of compact Hausdorff spaces?

Note added in the light of @Tyrone's comment. The answer to this question is NO as can be seen by consider the directed colimit of maps $S^1\to S^1\to S^1\to\dots$ in which the $n$th map is defined $z\mapsto x^{n!}$. The colimit contains a point (namely the image of $1\in S^1$) whose preimage is dense and so the colimit fails the weaker separation axiom $T_1$.

In the light of this I should broaden the question: what I am really interested in is finding attractive ways of drawing attention to the weakly Hausdorff property and how it sits in relation to the world of compact Hausdorff spaces.

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    $\begingroup$ Filtered colimits of compact Hausdorff spaces need not be weakly Hausdorff. See Ex. 2 on pg.422 of Dugundji's Topology book. Maybe you want to be clearer about which colimits you wish to allow and where they should be computed? $\endgroup$
    – Tyrone
    Commented Aug 8, 2022 at 12:01
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    $\begingroup$ The correct characterization is that they are the topological spaces that can be written as filtered colimits of compact Hausdorff spaces along injective transition maps. It is pretty clear that all CGWH spaces are of this form (being CG, it is the filtered colimit of the images of maps from CH spaces; and those images are themselves CH by WH); the other direction is e.g. Proposition A.14 in Schwede's "Global homotopy theory". $\endgroup$
    – Peter Scholze
    Commented Aug 9, 2022 at 9:46
  • $\begingroup$ Compactly generated spaces can be described in terms of the lifting property as $\bigcup(\{\{0\leftrightarrow 1\}\to\{0=1\}\}\cup\{\varnothing \to K \,\,:\,\, K\,\, \text{ compact}\}\big)^{rl}$, see details of the notation at ncatlab.org/nlab/show/lift#ExamplesOfLiftingProperties $\endgroup$
    – user420620
    Commented Sep 5, 2022 at 15:11

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Peter Scholze's comment

The correct characterization is that they are the topological spaces that can be written as filtered colimits of compact Hausdorff spaces along injective transition maps. It is pretty clear that all CGWH spaces are of this form (being CG, it is the filtered colimit of the images of maps from CH spaces; and those images are themselves CH by WH); the other direction is e.g. Proposition A.14 in Schwede's "Global homotopy theory".

is the correct answer to my question: thank you for this!

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