An overt space is a space that satisfies a condition logically dual to that satisfied by a compact space. An overt space is also called open (in French, there is only one word, ‘ouvert’).
Recall that a topological space $X$ is compact if and only if, for every other space $Y$ and any open subset $U$ of $X \times Y$, the subset
is open in $Y$.
Dually, a space is overt if and only if, for every other space $Y$ and any open subset $U$ of $X \times Y$, the subset
is open in $Y$. (Note that the duality here is only in the logical connectives, not within the category of spaces.)
Of course, every topological space satisfies this condition; $\exists_X U$ is the union over $a$ of the open subsets $\{ b \;|\; (a,b) \in U \}$. However, the condition is not trivial in some frameworks, such as constructive locale theory, formal topology, and Abstract Stone Duality.
To remove it from dependence on points, we write the definition like this:
A space (topological space, locale, etc) $X$ is overt (or open) if, given any space $Y$ and any open subset $U$ of the product space $X \times Y$, there exists an open $\exists_X U$ in $Y$ that satisfies the universal property of existential quantification:
for every open $V$ in $Y$.
Note that, since we quantify over all spaces $Y$ and opens $U\subseteq X\times Y$ in this definition, whether a given space is overt may depend on precisely what ‘space’ and ‘open’ mean (even if $X$ is an example for both meanings). For example, if $X$ is a set with the discrete topology, then it is always overt if ‘space’ means ‘topological space’ or ‘locale’ with the usual meaning of ‘open’. On the other hand, if ‘space’ means simply ‘set’, but ‘open’ refers to the synthetic notion induced by a dominance, then overtness of $X$ is a nontrivial condition (and in fact, if all sets are overt in this sense then the dominance is trivial).
In the case of locales, it is sufficient to require the above definition only for the case $Y \coloneqq 1$ the point (unlike the “dual” case of compactness). In this case, when the map $\exists_X\colon Op(X) \to Op(1) = TV$ (the set or class of truth values) exists, it is a positivity predicate on $Op(X)$. The behavior of this predicate depends on the foundational assumptions:
In classical mathematics, it always exists; thus classically every locale is overt.
In impredicative constructive mathematics (such as the internal logic of a topos), the positivity predicate can be defined, but it may not satisfy the requisite univeral property of adjointness. Thus, constructively, not every locale is overt. However, even constructively, every topological locale is overt (so a sober space is overt regardless of whether it is viewed as a topological space or as a locale).
In constructive predicative mathematics, a positivity predicate cannot be defined and so must be given as a structure on (predicative data that generate) the locale, as is done with a formal topology. Once this structure is assumed, one can then ask whether a formal topology is overt, i.e. whether the axiomatic positivity predicate satisfies the requisite adjointness.
In impredicative constructive mathematics, overt locales can be characterized by the positive covering lemma: $X$ is overt iff every open $U\in O(X)$ can be covered by positive opens, and iff every covering of an open $U$ can be refined to a covering by positive elements.
At least in impredicative constructive mathematics, the locale induced by a topological space is overt. An open subset $U$ is positive if and only if it is inhabited.
At least in impredicative constructive mathematics, the spectrum of a commutative ring $A$ (defined as the locale whose frame is the frame of radical ideals) is overt if and only if any element of $A$ is nilpotent or not nilpotent.
As compact spaces go with proper maps, so overt spaces go with open maps. Indeed, $X$ is compact if and only if the unique map $X \to pt$ to the point is proper, while $X$ is overt if and only if the unique map $X \to pt$ is open. Similarly, if $X\to pt$ is instead closed, then $X$ is covert.
Note that overtness is expressed only in terms of a left adjoint, whereas open maps of locales must additionally satisfy a Frobenius reciprocity condition. In the case of locale maps to the point, this latter condition is automatic.
An overt space may also be called locally $(-1)$-connected, since this condition is the (0,1)-topos-theoretic version of the notion of locally connected topos and locally n-connected (n,1)-topos. A similar thing happens for higher local connectivity involving the Frobenius reciprocity condition, which must be imposed on general geometric morphisms to make them locally $n$-connected, but is automatic for morphisms to the point.
In synthetic topology, we interpret ‘space’ to mean simply ‘set’ (or type, i.e. the basic objects of our foundational system). If the notion of ‘open’ is specified by a dominance, then there is an induced nontrivial notion of “overt set”, defined essentially as above. For instance, the Rosolini dominance is the smallest dominance such that $\mathbb{N}$ is overt, whereas if all sets are overt then the dominance is trivial.
On the other hand, if by ‘open’ we mean an open subset in the sense of Penon, then all sets are overt.
The notion of an open locale was originally introduced by Joyal and Tierney (and developed by Johnstone in Stone Spaces):
The term “overt” is due to Paul Taylor. For example, it appears in:
Some of the history is described in the introduction to: