Bi-topological spaces and the Continuity Problem

TL;DR

This paper investigates the continuity of effective operators within bi-topological spaces, extending previous topological frameworks and establishing conditions under which such operators are continuous or effective, with applications to quasi-metric spaces.

Contribution

It extends the topological setting to bi-topological spaces and proves new conditions for the continuity and effectivity of operators, including in quasi-metric spaces.

Findings

Effective operators are continuous if the codomain is effectively regular.

Bi-continuous operators are shown to be effective.

Quasi-metric spaces satisfy the effectivity conditions under reasonable computability assumptions.

Abstract

The \emph{Continuity Problem} is the question whether effective operators are continuous, where an effective operator $F$ is a function on a space of constructively given objects $x$, defined by mapping construction instructions for $x$ to instructions for $F(x)$ in a computable way. In the present paper the problem is dealt with in a bi-topological setting. To this end the topological setting developed by the author \cite{sp} is extended to the bi-topological case. Under very natural conditions it is shown that an effective operator $F$ between bi-topological spaces $\TTT = (T, \tau, \sigma)$ and $\TTT' = (T', \tau', \sigma')$ is (effectively) continuous, if $\tau'$ is (effectively) regular with respect to $\sigma'$. A central requirement on $\TTT'$ is that bases of the neighbourhood filters of the points in $T'$ can computably be enumerated in a uniform way, not only with respect to topology $\tau'$, but also with respect to $\sigma'$. As follows from an example by Friedberg, the last condition is indispensable. Conversely, it is proved that (effectively) bi-continuous operators are effective. A prominent example of bi-topological spaces are quasi-metric spaces. Under a very reasonable computability requirement on the quasi-metric it is shown that all effectivity assumptions made in the general results are satisfied in the quasi-metric case.