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Is there an elementary way to define Haussdorf-compactly generated weakly Hausdorff topological spaces in a way that does not need defining topological space first?

Weakly Hausdorff sequential spaces have an alternative cryptomorphic axiomatization as Hausdorff subsequential spaces, which are arguably simpler to define than topological spaces, and have the property that dropping the Hausdorff condition turns it into a quasitopos.

One would hope then that compactly generated k-Hausdorff spaces have a similarly well behaved definition that can skip the usual presentation of things, since the category is very natural. Is there such a presentation?

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    $\begingroup$ Could you explain how defining `Hausdorff subsequential spaces' is simpler than defining topological spaces? $\endgroup$
    – Tyrone
    Commented Mar 12, 2023 at 13:52
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    $\begingroup$ @Tyrone because morphisms of sequential spaces are characterized as ones preserving an (infinitary) operation of a limit; whereas continuous maps cannot be defined without referring to elements and involve a very ill-behaved notion of preimage, or an even nastier notion of induced mapping on set of subests. $\endgroup$
    – Denis T
    Commented Mar 13, 2023 at 12:49
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    $\begingroup$ @DenisT that’s not simpler. $\endgroup$ Commented Mar 13, 2023 at 14:54
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    $\begingroup$ @FernandoMuro it is "arguably" simpler as in, "there are situations where this is simpler" as explained by Denis T. Some results, for e.g. Manes theorem or Tychonov theorem are much more natural when you think in terms of limits as an operation, instead of open sets. One could also argue that most students encounter the notion of limits long before the notion of open sets, so maybe axiomatizing the notion of limits is more natural than axiomatizing the notion of open sets. Not that I'm not saying this is the case - this is obviously opinion based, but you can't deny this can be argued. $\endgroup$ Commented Mar 13, 2023 at 18:07
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    $\begingroup$ One way to view it as simpler is that topological spaces formulated in terms of filters or nets are much more complex than subsequential/sequential spaces formulated in terms of sequences. The open set formulation of topological spaces is simple, but the open set formulation of topological spaces with nice properties isn't usually as simple and you often end up with an ugly category unless you add a bunch of seemingly arbitrary conditions $\endgroup$
    – saolof
    Commented Mar 13, 2023 at 18:39

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Sorry, I misremembered this. There are more quasiseparated condensed sets than just the weak Hausdorff $k$-spaces. See p. 15, right after Prop 2.9 here. I don't know if there's a way to define weak Hausdorff $k$-spaces in terms of condensed sets alone.

Weakly Hausdorff k-spaces are the quasiseparated objects in the category of condensed sets (see Thm 2.16 here). Quasiseparated is a categorical notion which is standard in topos theory.

In turn, the category of condensed sets can be defined as

  • sheaves on compact Hausdorff spaces (= algebras for the ultrafilter monad on $Set$ = opposite category of commutative unital $C^\ast$ algebras), or as
  • sheaves on totally disconnected compact Hausdorff spaces (= profinite sets), or as
  • sheaves on extremally disconnected spaces (=idempotent completion of the Kleisli category for the ultrafilter monad), or as
  • sheaves on free extremally disconnected spaces (=the Kleisli category for the ultrafilter monad).

Here, the ultrafilter monad is the unique monad whose underlying functor is the ultrafilter functor $\beta : Set \to Set$, carrying a set to the set of ultrafilters thereon. The topology with respect to which we take sheaves is some additional data in the first two descriptions, but in the last two descriptions it's very straightforward: a sheaf is a presheaf carrying coproducts to products.

Anyway, that gives a few possible definitions which don't require you to know what a topological space is.

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  • $\begingroup$ I've never encountered this "categorical notion which is standard in topos theory". (-: Can you give a reference? $\endgroup$ Commented Mar 16, 2023 at 3:07
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    $\begingroup$ @MikeShulman I think it should be in SGA4-VII somewhere. Definition is an obvious one mimicking situation for T1 spaces: one says that an object of a topos is quasicompact if every covering (effective epi from coproduct) admits a finite subcovering, and X being quasiseparated means that (binary) fibered products over X preserve quasicompactness. $\endgroup$
    – Denis T
    Commented Mar 16, 2023 at 13:03
  • $\begingroup$ @MikeShulman Denis T has given the definition; I know I first encountered the concept when reading about how to recover a coherent pretopos from a coherent topos, since a coherent object is one which is quasicompact and quasiseparated. I just looked in Makkai and Reyes Def 9.2.1 and they don't give a name to quasiseparatedness though. But presumably it is defined in SGA. $\endgroup$
    – Tim Campion
    Commented Mar 16, 2023 at 14:13
  • $\begingroup$ Ah, this sounds like the notion that is called stable in D3.3 of the Elephant? And since non-algebraic-geometers generally drop the "quasi" from "compact", is it also common to drop it from "separated"? $\endgroup$ Commented Mar 16, 2023 at 19:45

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