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The Journal of Symbolic Logic
Journal Article

On Effective Topological Spaces

Dieter Spreen
The Journal of Symbolic Logic
Vol. 63, No. 1 (Mar., 1998), pp. 185-221
Published by: Association for Symbolic Logic
DOI: 10.2307/2586596
https://www.jstor.org/stable/2586596
Page Count: 37

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Topics: Numberings, Topological theorems, Index sets, Natural numbers, Logical theorems, Separable spaces, Recursion
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On Effective Topological Spaces
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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 Ceĭtin-Moschovakis, and a result by Eršov and Berger which says that the hereditarily effective operations coincide with the hereditarily effective total continuous functionals on the natural numbers.
The Journal of Symbolic Logic © 1998 Association for Symbolic Logic