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 $X$ is compact if and only if, for every other space $Y$ and any open subset $U$ of $X \times Y$, the subset

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

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

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

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.

Definition

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 $U$ in the product space $X \times Y$, there exists an open $\exists_X U$ in $Y$ that satisfies the universal property of existential quantification:

$\exists_X U \subseteq V \;\Leftrightarrow\; U \subseteq X \times V$

for every open $V$ in $Y$.

Note that, since we quantify over all spaces $Y$ in this definition, whether a given space is overt may depend on precisely what ‘space’ means (even if $X$ 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 $Y \coloneqq 1$ 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 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, $X$ 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.

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.

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