Bi-topological spaces and the Continuity Problem - Spreen, Dieter (Topology Proceedings, 2021)
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 T=(T, τ, σ) and T^′=(T^′, τ^′, σ^′) is (effectively) continuous, if τ^′ is (effectively) regular with respect to σ^′. A central requirement on T^′ 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 τ^′, but also with respect to σ^′. 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.