In synthetic topology done as a branch of constructive mathematics, a dominance is a set $\Sigma$ that functions as an analogue of the Sierpinski space. In particular, it allows us to define synthetically a notion of open subset: $\Sigma$ is a subset of the set of truth values $\Omega$, and a subset of a set $A$ is called “open” if its classifying map $A\to \Omega$ lands in $\Sigma$.
The name dominance is meant to evoke that the set is used to define the domains of a class of partial functions. I.e., in synthetic topology the partial functions whose domain is an open set and in synthetic computability theory the domains of partial computable functions.
Let $\Omega$ be the set of truth values. A dominance is a subset $\Sigma\subseteq \Omega$ such that
The second condition implies that $\Sigma$ is closed under binary meets $\wedge$, and hence is a sub-meet-semilattice of $\Omega$. In type-theoretic language, the second condition says that $\Sigma$ is closed under dependent sums.
The elements of $\Sigma$ are called open truth values.
In homotopy type theory, every univalent type of propositions is a subtype of the type of all propositions $\Omega$. A dominance is a univalent type of propositions $(\Sigma, \mathrm{El}_\Sigma)$ with an element $\top:\Sigma$ such that
We define a subset $U\subseteq A$ of an arbitrary set $A$ to be open if for each $x\in A$, the proposition “$x\in U$” is an open truth value. The second condition above is equivalent to saying that if $U\subseteq A$ is open and also $V\subseteq U$ is open, then $V\subseteq A$ is open.
Note that for any function $f:A\to B$, the preimage of any open set is open, since $(x\in f^{-1}(U))\iff (f(x) \in U)$. Thus, any function is “continuous” with respect to this “intrinsic topology.”
It is hard to get very far without an additional assumption that $\Sigma$ is closed under some joins as well. However, if it were closed under all joins, then it would be all of $\Omega$, since any $P\in \Omega$ is the join $\bigvee_{i\in \{\star | P\}} \top$.
Given a dominance $\Sigma$, we say that a set $I$ is overt if $\Sigma$ is closed under $I$-indexed joins. (This is related to, but not identical to, the notion of overt space.) In general it is reasonable to expect discrete sets to be overt in this sense. In some frameworks such as spatial type theory there is a formal notion of “discrete” and we can actually assert that all discrete sets are overt. Otherwise we can assume that specific sets that we expect to be discrete are overt. For instance, we might assume that:
The singleton $\{\top\}$ is a dominance, for which only singletons are overt.
The set $\{\bot,\top\}$ is a dominance. This is the smallest dominance such that the empty set is overt. (In classical mathematics, of course, this and the previous example are the only two dominances, and the theory trivializes.)
The set of all truth values of the form $\exists n, f(n) = 1$ for some function $f:\mathbb{N}\to \mathbf{2}$ is often a dominance, though this may not be provable without further assumptions. For instance, this is the case if we assume countable choice or (perhaps) the propositional axiom of choice. When it is a dominance, this is the smallest dominance such that $\mathbb{N}$ is overt; it is called the Rosolini dominance. Equivalently, it is the set of truth values of the form $x\gt 0$ for some Cauchy real number $x$.
The set of all truth values of the form $x\gt 0$ for some Dedekind real number $x$ is also often a dominance, though this also may not be provable without further assumptions.
G. Rosolini. Continuity and Effectiveness in Topoi. PhD thesis, University of Oxford, 1986. pdf
Martin Escardo, Topology via higher-order intuitionistic logic., unfinished paper, pdf
Martin Escardo, Synthetic topology of data types and classical spaces. Electronic Notes in Theoretical Computer Science, 87:21–156, 2004. pdf
Andrej Bauer and Davorin Lesnik, Metric Spaces in Synthetic Topology, pdf
Last revised on February 10, 2023 at 20:35:05. See the history of this page for a list of all contributions to it.