Some results related to the continuity problem

TL;DR

This paper investigates the effective continuity of maps between topological spaces, showing that under certain conditions, effective operators are effectively sequentially continuous, with implications for the classical continuity theorem and the Axiom of Choice.

Contribution

It demonstrates that the witness condition for effective maps naturally arises from the classical proof involving the Axiom of Choice, for specific classes of spaces.

Findings

Effective maps with witnesses are effectively continuous in certain classes of spaces.

The witness condition is linked to the use of the Axiom of Choice in classical proofs.

The general case remains an open question.

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 noninclusion. 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 $\textit{not}$ required to be Hausdorff. They only need to satisfy the weaker $T_0$ separation condition.