This question is set in constructive mathematics (without Choice), such as in the internal logic of a topos with natural numbers object, or in IZF.
Short version of the question: if $X$ is a sober topological space and $\newcommand{\Loc}{\operatorname{Loc}}\Loc(X)$ the associated locale, then open subsets of $X$ and open sublocales of $\Loc(X)$ can be identified in a straightforward way; classically, this is also the case of closed subsets and closed sublocales; however, constructively, if we take the most natural definition of “closed” (namely that all adherent points belong to the subset, “adh-closed” below), then the correspondence between closed subsets of $X$ and closed sublocales of $\Loc(X)$ breaks down. Can we repair this on the localic side?
Longer version of the question and discussion follow.
The following definitions and facts should be standard (at least classically), but are recalled so we're all on the same page (so skip unless in doubt):
A frame is a lattice admitting arbitrary joins ($\bigvee$) and such that finite meets ($\wedge$) distribute over these. Frame homomorphisms are maps preserving finite meets (including the top element) and arbitrary joins, and a subframe is a subset such that the inclusion is a frame homomorphism. The Heyting operation on a frame is defined by $(U\Rightarrow W) := \bigvee_{(U\wedge V)\leq W} V$.
A nucleus on a frame $L$ is a map $j\colon L\to L$ which preserves finite meets and satisfies $U\leq j(U)$ and $j(j(U)) = j(U)$ for all $U\in L$. The quotient of $L$ by the equivalence relation $U \sim U'$ when $j(U) = j(U')$ is then a frame in the obvious way, and conversely any quotient frame arises in this way through a unique nucleus (which associates to each equivalence class the greatest element of the class).
A locale is an object in the opposite category to that of frames. We write $\mathcal{O}(X)$ for the frame associated to a locale $X$ and we call it its “frame of opens”. We write $f^*\colon \mathcal{O}(Y) \to \mathcal{O}(X)$ for the morphism of frames associated to a morphism of locales $f\colon X\to Y$. A sublocale is a morphism of locales such that the associated morphism of frames is a quotient map (=surjection), so it can be defined by a (uniquely defined) nucleus. For $U\in\mathcal{O}(X)$, the open sublocale$\mathfrak{o}_U$ defined by $U$ is the one associated to the nucleus $W \mapsto (U\Rightarrow W)$ (its frame of opens is $\mathcal{O}(U) := \{V \in \mathcal{O}(X) : V\leq U\}$ with $U$ as top element, and binary meets and arbitrary joins as in $\mathcal{O}(X)$), and we can identify this locale with $U$; and the (complementary) closed sublocale$\mathfrak{c}_U$ is the one associated to the nucleus $W \mapsto U\vee W$.
A topological space is a set $X$ endowed with a subframe $\mathcal{O}(X)$ of its powerset $\mathcal{P}(X)$, called the “frame of opens” of $X$, or its “topology”. (Elements of $\mathcal{O}(X)$ are, of course, called “open sets”, or “open neighborhoods” of any of their points.) A morphism (=continuous map) $f\colon X\to Y$ between topological spaces is a map of the underlying sets such that the inverse image map $f^{-1}\colon \mathcal{P}(Y)\to \mathcal{P}(X)$ takes open sets to open sets, i.e., restricts to a map (so a morphism of frames) $f^*\colon \mathcal{O}(Y) \to \mathcal{O}(X)$ of the frames of opens. Consequently we get an obvious “associated locale” functor from the category of topological spaces to that of locales (taking $X$ to the locale $\Loc(X)$ associated to the frame $\mathcal{O}(X)$). This functor has a right adjoint, the “space of points” functor which takes a locale $X$ to the topological space whose underlying set is the set $\newcommand{\Pts}{\operatorname{Pts}}\Pts(X)$ of morphisms $1\to X$ (where $1$ is the terminal locale, the one whose frame is the set $\Omega := \mathcal{P}(1)$ of truth values), with topology defined by the image of the frame morphism $\mathcal{O}(X) \to \mathcal{P}(\Pts(X))$ given by $U \mapsto \Pts(U)$. The adhunction is idempotent. The topological space $X$ is called sober when the unit $\eta_X \colon X \to \Pts(\Loc(X))$ of this adjunction is a homeomorphism (viꝫ. an isomorphism of topological spaces), i.e., when $X$ is of the form $\Pts(—)$, and it is then legitimate to identify $X$ with $\Loc(X)$. Conversely, the locale $X$ is called spatial (or said to “have enough points”) when the coünit $\varepsilon_X \colon \Loc(\Pts(X)) \to X$ is an isomorphims of locales, i.e., when $X$ is of the form $\Loc(—)$.
To any subset $E\subseteq X$ of a topological space we can associate a sublocale $\mathfrak{s}_E$ of $\Loc(X)$ (the “subspace sublocale”) defined by the nucleus which takes $W$ to the largest open set $\bigcup_{(E\cap V)\subseteq W} V$ whose intersection with $E$ is included in $W$: as a locale, it is the spatial locale associated to $E$ (with the subspace topology). If $E$ is open, this is just the open sublocale $\mathfrak{o}_E$ mentioned above.
The following definitions are not standard, however. Classically, a “closed set” is defined as the complement of an open set or equivalently, as one which is equal to its adherence; constructively, these are different notions:
If $X$ is a topological space, we say that $x\in X$ is adherent to a subset $F\subseteq X$ when for every open neighborhood of $x$ the intersection $F\cap U$ is inhabited. We say that $F$ is adh-closed when every $x\in X$ that is adherent to $F$ belongs to $F$.
If $X$ is a topological space and $U \in \mathcal{O}(X)$, we let $(X\setminus U) := \{x\in X : \neg(x\in U)\}$. We say that $F$ is cpl-closed if there exists $U \in \mathcal{O}(X)$ such that $F = (X\setminus U)$. Note that if this is the case, then some such $U$ (but not necessarily the one we stared with!) can be reconstructed as the set $F^\#$ of $x\in X$ such that there is an open neighborhood $V$ of $x$ with $F\cap V = \varnothing$ (more simply: the largest open set disjoint from $F$). In other words, to say that $F$ is cpl-closed means that every $x\in X$ such that for every open neighborhood $V$ of $x$ the intersection $F\cap V$ is not empty (“not empty” means just $\neg(F\cap V = \varnothing)$) belongs to $F$.
Clearly, “cpl-closed” implies “adh-closed”. A Brouwerian counterexample to the converse: if $p \in \Omega$ is a truth value, and $X$ an arbitrary topological space, the subset $\{x\in X : p\}$ is adh-closed, but to say it is cpl-closed means $\neg\neg p$ implies $p$.
(Arguably, “adh-closed” is the better behaved notion, and deserves to be called “closed”. But in the context of this question, I thought it preferable to be more explicit. Also note that both notions might fail to be stable under binary unions, e.g. $[-1,0]$ and $[0,1]$ in the Dedekind reals are both cpl-closed, but the adherence of their union is $[-1,1]$, so the union is adh-closed iff it holds that every real is $\geq 0$ or $\leq 0$.)
Open sets are neatly represented at the localic level: by definition, if $X$ is a topological space, the set $\mathcal{O}(X)$ of open subsets of $X$ can be identified with that of open sublocales of $\Loc(X)$, and said sublocales are all spatial. If $X$ is sober, the points of the open sublocale $\mathfrak{o}_U$ are the same as the points of the open set $U$, and we can thoughtlessly identify open sets and open sublocales.
Classically, the situation for closed sets is just as neat: a closed set $F \subseteq X$ is the complement of an open set $U$, and when $X$ is sober this is also the space of points of the complementary closed sublocale $\mathfrak{c}_U$; furthermore, this sublocale is spatial and indeed coincides with the subspace sublocale $\mathfrak{s}_F$.
Constructively, however, the situation for closed sets is far less felicitous. If $U$ is an open set of a sober topological space $X$, the set of points of the complementary closed sublocale $\mathfrak{c}_U$ defined by $U$ is $X \setminus U$, so, a cpl-closed set: so closed sublocales (might) fail to represent the more interesting notion of adh-closed sets. Also, even when $F = X \setminus U$ is cpl-closed, the subspace sublocale $\mathfrak{s}_F$ it defines need not coincide with the closed complementary closed sublocale $\mathfrak{c}_U$ defined by $U$— the latter need not be spatial. In fact, that $X \setminus U = X \setminus U'$ need not imply $\mathfrak{c}_U = \mathfrak{c}_{U'}$.
(Again, to give an example of what can go wrong, let $p$ be a truth value, assume $p$ is regular, that is $p \Leftrightarrow \neg\neg p$, and consider the cpl-closed set $F := \{x\in X : p\}$, which is also open, and which is the complement of the open set $U := \{x\in X : \neg p\} = F^\#$. Yet $\mathfrak{s}_F = \mathfrak{o}_F$ takes $F$ to the top element whereas $\mathfrak{c}_U$ takes $F$ to $\{x\in X : p\lor\neg p\}$.)
Questions: Basically I want to ask whether we can find a way to “fix this mess” and represent the adh-closed subsets of a sober space $X$ as sublocales of $\Loc(X)$. So:
Can we characterize the subspace sublocales $\mathfrak{s}_F$ defined by the adh-closed subsets $F \subseteq X$?
Can we find a map associating to each adh-closed subset $F \subseteq X$ a sublocale, say $\mathfrak{a}_F$, of $\Loc(X)$, such that $\mathfrak{a}_F \leq \mathfrak{a}_{F'}$ (as sublocales; this means the reverse inequality on the nuclei) iff $F \subseteq F'$ as subsets? Can we additionally require that $\mathfrak{a}_F = \mathfrak{c}_U$ when $F$ is cpl-closed with $F = X\setminus U$ and $U = F^\#$?
(Feel free to interpret the above questions liberally and propose other ways to represent subsets of $X$ on the localic side. However, saying “forget about point-set topology and just look at the locales” does not count as an answer: I am really interested in things such as “the reals such that $0\leq x\leq 1$ and $p$”.)
