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Recursive constructions in topological spaces

Published online by Cambridge University Press:  12 March 2014

Iraj Kalantari  and
Allen Retzlaff
Iraj Kalantari
Affiliation:
University of California, Santa Barbara, California 93106
Allen Retzlaff
Affiliation:
Western Illinois University, Macomb, Illinois 61455 State University of New York, Purchase, New York 10577
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Abstract

We study topological constructions in the recursion theoretic framework of the lattice of recursively enumerable open subsets of a topological space X. Various constructions produce complemented recursively enumerable open sets with additional recursion theoretic properties, as well as noncomplemented open sets. In contrast to techniques in classical topology, we construct a disjoint recursively enumerable collection of basic open sets which cannot be extended to a recursively enumerable disjoint collection of basic open sets whose union is dense in X.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 44 , Issue 4 , December 1979 , pp. 609 - 625
Copyright
Copyright © Association for Symbolic Logic 1979

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References

REFERENCES

[1]Birkhoff, G., Lattice theory, American Mathematical Society Colloquium Publications, vol. 25, Providence, R. I., 1967.Google Scholar
[2]Frölich, A. and Shepherdson, J. C., Effective procedures in field theory, Philosophical Transactions of the Royal Society of London, Series A, vol. 284(1955), pp. 407–432.Google Scholar
[3]Kalantari, I., Automophisms of the lattice of recursively enumerable vector spaces, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 25 (1979).CrossRefGoogle Scholar
[4]Kalantari, I., Major subspaces of recursively enumerable vector spaces, this Journal, vol. 43(1978), pp. 293–303.Google Scholar
[5]Kalantari, I. and Retzlaff, A., Maximal vector spaces under automorphisms of the lattice of recursively enumerable vector spaces, this Journal, vol. 42(1977), pp, 481–491.Google Scholar
[6]Kalantari, I. and Retzlaff, A., Degrees of recursively enumerable topological spaces (forthcoming).Google Scholar
[7]Kelley, J., General topology, Van Nostrand, Princeton, NJ, 1955.Google Scholar
[8]Lachlan, A. H., On the lattice of recursively enumerable sets, Transactions of the American Mathematical Society, vol. 130(1968), pp. 1–37.CrossRefGoogle Scholar
[9]Lacombe, D., Les ensembles récursivement ouverts ou fermés, et leurs applications à l'analyse récursive, Comptes Rendus Hebdomadaires des Séances del'Académie des Sciences (Paris), vol. 245(1957), pp. 1040–1043; D. Lacombe, Les ensembles récursivement ouverts ou fermés, et leurs applications à l'analyse récursive, Comptes Rendus Hebdomadaires des Séances del'Académie des Sciences (Paris), vol. 245(1957) vol. 246 (1958), pp. 28–31.Google Scholar
[10]Lacombe, D., Quelques procédés de definition en topologie récursive, Constructivity in mathematics, (Heyting, A., Editor), North-Holland, Amsterdam, 1959, pp. 129–158.Google Scholar
[11]Metakides, G. and Nerode, A., Recursion theory and algebra, Algebra and Logic, Lecture Notes in Mathematics, vol. 450, Springer-Verlag, Berlin and New York, 1975, pp. 209–219.CrossRefGoogle Scholar
[12]Metakides, G. and Nerode, A., Recursively enumerable vector spaces, Annals of Mathematical Logic, vol. 11(1977), pp. 147–171.CrossRefGoogle Scholar
[13]Moschovakis, Y.N., Recursive metric spaces, Fundamenta Mathematicae, vol. 55(1964), pp. 215–238.CrossRefGoogle Scholar
[14]Moschovakis, Y.N., Notation systems and recursive ordered fields, Compositio Mathematica, vol. 17(1965), pp. 40–71.Google Scholar
[15]Post, E. L., Recursively enumerable sets of positive integers and their decision problems, Bulletin of the American Mathematical Society, vol. 50(1944), pp. 284–316.CrossRefGoogle Scholar
[16]Rabin, M. O., Computable algebra, general theory and theory of computable fields, Transactions of the American Mathematical Society, vol. 95(1960), pp. 341–360.Google Scholar
[17]Remmel, J., On r.e. and ω-r.e. vector spaces with nonextendible bases, this Journal (to appear).Google Scholar
[18]Retzlaff, A., Simple and hyperhypersimple vector spaces, this Journal, vol. 43(1978), pp. 260–269.Google Scholar
[19]Retzlaff, A., Direct summands of r.e. vector spaces, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik (to appear).Google Scholar
[20]Rogers, H., Theory of recursive functions and effective computabiliiy, McGraw-Hill, New York, 1968.Google Scholar
[21]Sacks, G. E., Degrees of unsolvability(revised edition), Annals of Mathematical Studies, no. 55, Princeton University Press, Princeton, NJ, 1966.Google Scholar
[22]Specker, E., Der Satz vom Maximum in der rekursiven Analysis, Constructivity in mathematics (Heyting, A., Editor), North-Holland, Amsterdam, 1959.Google Scholar