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Some results related to the continuity problem

Published online by Cambridge University Press:  13 June 2016

DIETER SPREEN
DIETER SPREEN*
Affiliation:
Department of Mathematics, University of Siegen, 57068 Siegen, Germany. Email: spreen@math.uni-siegen.de Department of Decision Sciences, University of South Africa, PO Box 392, Pretoria 0003, South Africa
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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

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 27 , Special Issue 8: Continuity, Computability, Constructivity: From Logic to Algorithms 2013 , December 2017 , pp. 1601 - 1624
Copyright
Copyright © Cambridge University Press 2016 

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