2016/02/29 by Dieter Spreen, DIETER SPREEN
Computer Science · Mathematics · #Advanced Banach Space Theory #Axiom #Axiom of choice #Calculus (dental) #Fixed Point Theorems Analysis #Optimization and Variational Analysis #Topological space #Topology (electrical circuits) #Witness #acm:03D45 #acm:03D78 #acm:03F60 #math.LO #msc:03D45 #msc:03D78 #msc:03F60
paper · pdf · doi:10.1017/s0960129516000190
23 pages
arxiv created 2016/03/01 · openalex publication_date 2016/06/13 · arxiv updated 2016/06/22 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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 T 0 separation condition