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Some results related to the continuity problem

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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

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Amadio, R.M. and Curien, P.-L. (1998). Domains and Lambda-Calculi, Cambridge University Press, Cambridge.
Blanck, J. (1997). Domain representability of metric spaces. Annals of Pure and Applied Logic 83 (3) 225247.
Ceĭtin, G.S. (1962). Algorithmic operators in constructive metric spaces. Trudy Mat. Inst. Steklov 67 295361; English transl., Amer. Math. Soc. Transl., ser. 2, 64 1–80.
Edalat, A. (1997). Domains for computation in mathematics, physics and exact real arithmetic. Bulletin of Symbolic Logic 3 (4) 401452.
Egli, H. and Constable, R.L. (1976). Computability concepts for programming language semantics. Theoretical Computer Science 2 (2) 133145.
Eršov, Ju. L. (1972) Computable functionals of finite type. Algebra i Logika 11 (4) 367437; English transl., Algebra and Logic 11 (4) 203–242.
Eršov, Ju. L. (1973). The theory of A-spaces. Algebra i Logika 12 (4) 369416; English transl., Algebra and Logic 12 (4) 209–232.
Eršov, Ju. L. (1975). Theorie der Numerierungen II. Zeitschrift für mathematische Logik Grundlagen der Mathematik 21 (1) 473584.
Eršov, Ju. L. (1977). Model ℂ of partial continuous functionals. In: Gandy, R. et al. (eds.) Logic Colloquium 76, North-Holland, Amsterdam, 455467.
Friedberg, R. (1958). Un contre-exemple relatif aux fonctionelles récursives. Comptes rendus hebdomadaires des séances de l'Académie des Sciences (Paris) 247 (1) 852854.
Gierz, G., Hofman, K.H., Keimel, K., Lawson, J.D., Mislove, M.W. and Scott, D. (2003). Continuous Lattices and Domains, Cambridge University Press, Cambridge.
Gunter, C.A. (1992). Semantics of Programming Languages, MIT Press, Cambridge, Mass.
Hoyrup, M. (2015) personal communication.
Kreisel, G., Lacombe, D. and Shoenfield, J. (1959). Partial recursive functionals and effective operations. In: Heyting, A. (ed.) Constructivity in Mathematics, North-Holland, Amsterdam, 290297.
Moschovakis, Y.N. (1964). Recursive metric spaces. Fundamenta Math. 55 (3) 215238.
Moschovakis, Y.N. (1965). Notation systems and recursive ordered fields. Compositio Math. 17 4071.
Myhill, J. and Shepherdson, J.C. (1955). Effective operators on partial recursive functions. Zeitschrift für mathematische Logik Grundlagen der Mathematik 1 (4) 310317.
Pour-El, M.B. and Richards, J.I. (1989). Computability in Analysis and Physics, Springer-Verlag, Berlin.
Rogers, H. Jr., (1967). Theory of Recursive Functions and Effective Computability. McGraw-Hill, New York.
Sciore, E. and Tang, A. (1978). Computability theory in admissible domains. In: 10th Annual ACM Symposium on Theory of Computing 95104. Association for Computing Machinery, New York.
Specker, E. (1949). Nicht konstruktiv beweisbare Sätze der Analysis. The Journal of Symbolic Logic 14 (3) 145158.
Spreen, D. (1995). On some decision problems in programming. Information and Computation 122 (1) 120139; Corrigendum (1999) 148 (2) 241–244.
Spreen, D. (1996). Effective inseparability in a topological setting. Annals of Pure and Applied Logic 80 (3) 257275.
Spreen, D. (1998). Effective topological spaces. The Journal of Symbolic Logic 63 (1) 185221.
Spreen, D. (2000). Corrigendum. The Journal of Symbolic Logic 65 (4) 19171918.
Spreen, D. (2001a). Can partial indexings be totalized? The Journal of Symbolic Logic 66 (3) 11571185.
Spreen, D. (2001b). Representations versus numberings: On two computability notions. Theoretical Computer Science 26 (1–2) 473499.
Spreen, D. (2008). On some problems in computable topology. In: Dimitracopoulos, C. et al. (eds.) Logic Colloquium'05, Cambridge University Press, Cambridge, 221254.
Spreen, D. (2010). Effectivity and effective continuity of multifunctions. The Journal of Symbolic Logic 75 (2) 602640.
Spreen, D. (2014). An isomorphism theorem for partial numberings. In: Brattka, V., Diener, H. & Spreen, D. (eds.) Logic, Computation, Hierarchies, Ontos Mathematical Logic 4, De Gruyter, Boston/Berlin, 341381.
Spreen, D. and Young, P. (1984). Effective operators in a topological setting. In: Richter, M.M., Börger, E., Oberschelp, W., Schinzel, B. & Thomas, W. (eds.) Computation and Proof Theory, Proceedings of the Logic Colloquium held in Aachen 1983, Part II. Lecture Notes in Mathematics 1104, Springer, Berlin, 437451.
Sturm, C.F. (1835). Mémoire sur la résolution des équations numeriques. Annales de mathématiques pures et appliquées 6 271318.
Weihrauch, K. and Deil, T. (1980). Berechenbarkeit auf cpo's. Schriften zur Angewandten Mathematik und Informatik, 63 Rheinisch-Westfälische Technische Hochschule Aachen.
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Mathematical Structures in Computer Science
  • ISSN: 0960-1295
  • EISSN: 1469-8072
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