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Simplicity in effective topology

Published online by Cambridge University Press:  12 March 2014

Iraj Kalantari  and
Anne Leggett
Iraj Kalantari
Affiliation:
Western Illinois University, Macomb, Illinois 61455
Anne Leggett
Affiliation:
Western Illinois University, Macomb, Illinois 61455
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Extract

The recursion-theoretic study of mathematical structures other than ω is now a field of much activity. Analysis and algebra are two such structures which have been studied for their effective contents. Studies in analysis began with the work of Specker on nonconstructive proofs in analysis [16] and in Lacombe's inspiring notes on relevant notions of recursive analysis [8]. Studies in algebra originated in the work of Frolich and Shepherdson on effective extensions of explicit fields [1] and in Rabin's study of computable fields [15]. Equipped with the richness of modern techniques in recursion theory, Metakides and Nerode [11]–[13] began investigating the effective content of vector spaces and fields; these studies have been extended by Kalantari, Remmel, Retzlaff, Shore and others.

Kalantari and Retzlaff [5] began a foundational inquiry into effectiveness in topological spaces. They consider a topological space X with a countable basis ⊿ for the topology. The space is fully effective, that is, the basis elements are coded into ω and the operation of intersection of basis elements and the relation of inclusion among them are both computable. Similar to , the lattice of recursively enumerable (r.e.) subsets of ω, the collection of r.e. open subsets of X forms a lattice ℒ(X) under the usual operations of union and intersection.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 47 , Issue 1 , March 1982 , pp. 169 - 183
Copyright
Copyright © Association for Symbolic Logic 1982

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References

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