Journal of Symbolic Logic

On Effective Topological Spaces

Dieter Spreen

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Abstract

Starting with D. Scott's work on the mathematical foundations of programming language semantics, interest in topology has grown up in theoretical computer science, under the slogan `open sets are semidecidable properties'. But whereas on effectively given Scott domains all such properties are also open, this is no longer true in general. In this paper a characterization of effectively given topological spaces is presented that says which semidecidable sets are open. This result has important consequences. Not only follows the classical Rice-Shapiro Theorem and its generalization to effectively given Scott domains, but also a recursion theoretic characterization of the canonical topology of effectively given metric spaces. Moreover, it implies some well known theorems on the effective continuity of effective operators such as P. Young and the author's general result which in its turn entails the theorems by Myhill-Shepherdson, Kreisel-Lacombe-Shoenfield and Ceitin-Moschovakis, and a result by Ersov and Berger which says that the hereditarily effective operations coincide with the hereditarily effective total continuous functionals on the natural numbers.

Article information

Source
J. Symbolic Logic Volume 63, Issue 1 (1998), 185-221.

Dates
First available: 6 July 2007

Permanent link to this document
http://projecteuclid.org/euclid.jsl/1183745466

JSTOR
links.jstor.org

Mathematical Reviews number (MathSciNet)
MR1610806

Zentralblatt MATH identifier
0915.03038

Citation

Spreen, Dieter. On Effective Topological Spaces. Journal of Symbolic Logic 63 (1998), no. 1, 185--221. http://projecteuclid.org/euclid.jsl/1183745466.


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See also

    Corrections

    • See Correction: Dieter Spreen. Corrigendum: On Effective Topological Spaces. J. Symbolic Logic, Volume 65, Issue 4 (2000), 1917--1918.