Recall that a topological space is compact if and only if, for every other space and any open subset of , the subset
is open in .
Dually, a space is overt if and only if, for every other space and any open subset of , the subset
is open in . (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; is the union over of the open subsets . 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) is overt (or open) if, given any space and any open subset of the product space , there exists an open in that satisfies the universal property of existential quantification:
for every open in .
Note that, since we quantify over all spaces and opens in this definition, whether a given space is overt may depend on precisely what ‘space’ and ‘open’ mean (even if is an example for both meanings). For example, if 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 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 the point (unlike the “dual” case of compactness). In this case, when the map (the set or class of truth values) exists, it is a positivity predicate on . 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: is overt iff every open can be covered by positive opens, and iff every covering of an open can be refined to a covering by positive elements.
As compact spaces go with proper maps, so overt spaces go with open maps. Indeed, is compact if and only if the unique map to the point is proper, while is overt if and only if the unique map is open. Similarly, if is instead closed, then 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 -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 -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 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 term “overt” is due to Paul Taylor. For example, it appears in:
Some of the history is described in the introduction to: