Bi-topological spaces and the Continuity Problem

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.