Some results related to the continuity problem

Abstract

The continuity problem, i.e., the question whether effective maps between effectively given topological spaces are effectively continuous, is reconsidered. In earlier work, it was shown that this is always the case, if the effective map also has a witness for non-inclusion. The extra condition does not have an obvious topological interpretation. As is shown in the present paper, it appears naturally where in the classical proof that sequentially continuous maps are continuous, the Axiom of Choice is used. The question is therefore whether the witness condition appears in the general continuity theorem only for this reason, i.e., whether effective operators are effectively sequentially continuous. For two large classes of spaces covering all important applications, it is shown that this is indeed the case. The general question, however, remains open.Spaces in this investigation are in general not required to be Hausdorff. They only need to satisfy the weaker T0 separation condition