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Some results related to the continuity problem
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 required to be
Hausdorff. They only need to satisfy the weaker separation condition.Comment: 23 page