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There are already questions on why a topological space is required to be closed under arbitrary unions. Summing up, it follows from other characterizations of topologies: from closure operators, neighborhoods, or "closeness" relations. Another comment recalls the usual argument where you choose a neighborhood of each point. However, all the reasons above rely heavily on points and closed sets.

Question. What are the reasons to require arbitrary joins in the case of locales, especially from a constructive point of view?

I know that, if we only require countable joins, then the right adjunction $\mathbf{Loc} \to \mathbf{Top}$ is going to fail. However, this seems hardly an argument if we are interested in the study of locales by themselves, with no reference to topological spaces. Below are some reasons why I find the study of locales closed only under countable joins (which are known as $\sigma$-locales, see the comments) compelling from a constructive point of view.

  • The requirement of closure under arbitrary joins is a source of impredicativity.
  • New interesting $\sigma$-locales appear. For example, I think the final $\sigma$-locale will be $\mathbf 2$ instead of the set of truth values $\Omega,$ and this allows for a distinction between "strong" points (from $\mathbf 2$) and "weak" points (from $\Omega$). I think the restriction that $\mathbf 2$ is not a locale is not necessary (except in some relations to logic).
  • More generally, one can define locales where joins are restricted to a particular index set, or a family of index sets.

The main arguments I find against this generalization is that perhaps some relation of locales to logic or formal topologies will be lost.

Edit. As noted in the comments, even though $\sigma$-frames have been studied, countable joins are still problematic from a predicative point of view. Let $F$ be a distributive lattice. It seems plausible to me to only allow joins over injections $\mathbf n \to F$ for each finite $\mathbf n$ and over injections $\mathbb N \to F$ (or, more generally, injections from sets from a fixed family). In this case, I hope that $\mathbf 2$ is really the initial $\sigma$-frame (final locale), which I find to be an interesting example.

Keplerto
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    I think they might be called $\sigma$-locales, by analogy with $\sigma$-algebras. – Zhen Lin Oct 14 '24 at 12:21
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    Sheaf toposes naturally contain a locale in their hearts (called "localic reflection"). But predicatively you do need to be careful with arbitrary unions, or even countable unions for that matter. This paper for example only allows finite unions, ie. distributive lattices. – Trebor Oct 14 '24 at 12:30
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    The final $\sigma$-locale is not 2. See for example HoTT book section 11.2 ("Dedekind reals") which includes a discussion on the initial $\sigma$-frame. – Trebor Oct 14 '24 at 12:35
  • I suggest to first find a class of interesting examples of c-locales (which are not locales). Otherwise the question has no "basis". – Martin Brandenburg Oct 14 '24 at 13:30
  • @Trebor I think you are right: if you define $U_n = \top$ if $P(n)$ and $U_n = \bot$ if $\lnot P(n),$ where $P(n)$ is decidable, then $\bigvee_n U_n = \top$ or $\bigvee_n U_n = \bot$ is equivalent to $(\exists n) P(n)$ or $(\forall n) \lnot P(n),$ which is non-constructive, so $2$ might not be a $\sigma$-frame. – Keplerto Oct 14 '24 at 14:24
  • On the other hand, the fact that $\sigma$-frames have a name indicate that they have nonetheless be studied, thanks @Zhen Lin for finding the name, because I did not. – Keplerto Oct 14 '24 at 14:26
  • Now I am not sure if I want $\sigma$-frames or distributive lattices as @Trebor suggested. Are they useful for topology aside from algebraic geometry? – Keplerto Oct 14 '24 at 14:26
  • Insisting on injective maps is problematic. Practically (especially considering examples) it is better to allow non-injective maps too. – Zhen Lin Oct 14 '24 at 23:04

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I have found the notion of a dominance. A dominance $D$ is a "simple" space (a version of the Sierpinski space). When we fix such a $D,$ it allows us to say that an index set $I$ is "valid" (overt) when $D$ is closed under $I$-indexed joins. Then one is to consider only those unions or joins over an overt index set.

When we take $D = \Omega$ we recover locales, whereas we can take $D = \Sigma$ closed under countable unions to recover $\sigma$-locales and related stuff. In one approach, $\Sigma$ is defined inductively instead of as a subset of $\Omega.$

What about the boolean domain $D = 2$? At first sight, only finite index sets would be "overt"; however, I think index sets can be more generally allowed to be compact, such as the cantor space. At least for the notion of compact here.

Two related subjects are Martin Escardo's theory of synthetic topology and Paul Taylor's abstract stone duality. I am interested in the foundational aspect of topology and I encourage those who know about these concepts to improve this answer.

Keplerto
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  • One of my motivations happens to be the Rice-Shapiro theorem, where only r.e. joins are needed. This computability motivation is shared with the above approaches. – Keplerto Oct 22 '24 at 17:21