nLab
overt space

Overt spaces

Idea

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’).

Motivation

Recall that a topological space XX is compact if and only if, for every other space YY and any open subset UU of X×YX \times Y, the subset

XU={b:Y|a:X,(a,b)U} \forall_X U = \{ b\colon Y \;|\; \forall\; a\colon X,\; (a, b) \in U \}

is open in YY.

Dually, a space is overt if and only if, for every other space YY and any open subset UU of X×YX \times Y, the subset

XU={b:Y|a:X,(a,b)U} \exists_X U = \{ b\colon Y \;|\; \exists\; a\colon X,\; (a, b) \in U \}

is open in YY. (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; XU\exists_X U is the union over aa of the open subsets {b|(a,b)U}\{ 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.

Definition

To remove it from dependence on points, we write the definition like this:

A space (topological space, locale, etc) XX is overt (or open) if, given any space YY and any open UU in the product space X×YX \times Y, there exists an open XU\exists_X U in YY that satisfies the universal property of existential quantification:

XUVUX×V \exists_X U \subseteq V \;\Leftrightarrow\; U \subseteq X \times V

for every open VV in YY.

Note that, since we quantify over all spaces YY in this definition, whether a given space is overt may depend on precisely what ‘space’ means (even if XX is an example for both meanings). I don't know any good examples of this, however.

Overt locales

In the case of locales, it is sufficient to require the above definition only for the case Y1Y \coloneqq 1 the point (unlike the “dual” case of compactness). In this case, when the map X:Op(X)Op(1)=TV\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)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 is determined uniquely if it exists, but it cannot be defined and so must be given axiomatically, 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.

Remarks

As compact spaces go with proper maps, so overt spaces go with open maps. Indeed, XX is compact if and only if the unique map XptX \to pt to the point is proper, while XX is overt if and only if the unique map XptX \to pt is open.

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)(-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 nn-connected, but is automatic for morphisms to the point.

Revised on February 28, 2011 23:15:26 by Toby Bartels (64.89.53.202)