Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-04T21:09:48.940Z Has data issue: false hasContentIssue false
  • English
  • Français

Everywhere Nonrecursive r.e. Sets in Recursively Presented Topological Spaces

Part of: Computability and recursion theory Generalities

Published online by Cambridge University Press:  09 April 2009

Li Xiang
Li Xiang
Affiliation:
Department of Mathematics Guizhou UniversityGuiyang People's Republic of China
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Recursively presented topological spaces are topological spaces with a recursive system of basic neighbourhoods. A recursively enumerable (r.e.) open set is a r.e. union of basic neighbourhoods. A set is everywhere r.e. open if its intersection with each basic neighbourhood is r.e. Similarly we define everywhere creative, everywhere simple, everywhere r.e. non-recursive sets and show that there exist sets both with and without these everywhere properties.

MSC classification

Secondary: 03D45: Theory of numerations, effectively presented structures 54A05: Topological spaces and generalizations (closure spaces, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 44 , Issue 1 , February 1988 , pp. 105 - 128
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Frohlich, A. and Shepherdson, J. C., ‘Effective procedures in field theory’, Phil. Trans. Roy. Soc. London Ser. A 248 (1955), 407–432.Google Scholar
[2]Hingston, P., Effectiveness in rings and topology (Ph.D. Dissertation, Monash University, 1983).Google Scholar
[3]Hingston, P., ‘Non-complemented open Sets in effective topologoy’, J. Austral. Math. Soc. (Series A), to appear.Google Scholar
[4]Kalantan, I. and Leggett, A., ‘Simplicity in effective topology’, J. Symbolic Logic 47 (1982), 169–183.CrossRefGoogle Scholar
[5]Kalantari, I. and Leggett, A., ‘Maximality in effective topology’, J. Symbolic Logic 48 (1983), 100–112.CrossRefGoogle Scholar
[6]Kalantari, I. and Remmel, J. B., ‘Degrees of unsolvability in effective topology’, J. Symbolic Logic 48 (1983), 610–622.CrossRefGoogle Scholar
[7]Kalantari, I. and Retzlaff, A., ‘Recursive constructions in topological spaces’, J. Symbolic Logic 44 (1979), 609–625.CrossRefGoogle Scholar
[8]Kalantari, I. and Weitkamp, G., ‘Effective topological spaces I: a definability theory’, Ann. Pure Appl. Logic 29 (1985), 1–27.CrossRefGoogle Scholar
[9]Kalantari, I. and Weitkamp, G., ‘Effective topolgical spaces II: a hierarchy’, Ann. Pure Appl. Logic 29 (1985), 207–224.CrossRefGoogle Scholar
[10]Kalantari, I. and Weitkamp, G., ‘Effective topological spaces III: forcing and definability’, Ann. Pure Appl. Logic, to appear.Google Scholar
[11]Metaiddes, G. and Nerode, A., Recursion theory and algebra, pp. 207–219 (Lecture Notes in Mathematics 450, Springer, 1975).Google Scholar
[12]Rabin, M. O., ‘Computable algebra, general theory and theory of computable fields’, Trans. Amer. Math. Soc. 95 (1960), 341–360.Google Scholar
[13]Rogers, H. Jr, Theory of recursive functions and effective computability (McGraw-Hill, 1967).Google Scholar
[14]Shoenfield, J. R., Degrees of unsolvability (North-Holland, 1971).Google Scholar