Bi-topological spaces and the Continuity Problem

Abstract

The 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 [22] is extended to the bi-topological case. Under very natural conditions it is shown that an effective operator $F$ between bi-topological spaces $\mathcal{T}=(T, \tau, \sigma)$ and $\mathcal{T}^{\prime}=\left(T^{\prime}, \tau^{\prime}, \sigma^{\prime}\right)$ is (effectively) continuous, if $\tau^{\prime}$ is (effectively) regular with respect to $\sigma^{\prime}$. A central requirement on $\mathcal{T}^{\prime}$ is that bases of the neighbourhood filters of the points in $T^{\prime}$ can computably be enumerated in a uniform way, not only with respect to topology $\tau^{\prime}$, but also with respect to $\sigma^{\prime}$. As follows from an example by Friedberg, the last condition is indispensable. Conversely, it is proved that (effectively) bi-continuous operators are effective. Prominent examples 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.