A Topological Analog to the Rice-Shapiro Index Theorem

Ever since Craig-Beth and Addison-Kleene proved their versions of the Lusin-Suslin theorem, work in model theory and recursion theory has demonstrated the value of classical descriptive set theory as a source of ideas and inspirations. During the sixties in particular, J.W. Addison refined the technique of "conjecture by analogy" and used it to generate a substantial number of results in both model theory and recursion theory (see, e.g., Addison [1], [2], [3]). During the past 15 years, techniques and results from recursion theory and model theory have played an important role in the development of descriptive set theory. (Moschovakis's book [6] is an excellent reference, particularly for the use of recursion-theoretic tools:,) The use of "conjecture by analogy" as a means of transferring ideas from model theory and recursion theory to descriptive set theory has developed more slowly. Some notable recent examples of this phenomenon are in Vaught [9], where some results in invariant descriptive set theory reflecting and extending model-theoretic results are obtained and others are left as conjectures (including a version of the well-known conjecture on the number of countable models) and in Hrbacek and Simpson [4], where a notion analogous to that of Turing reducibility is used to study Borel isomorphism types. Moschovakis [6] describes in detail an effective descriptive set theory based in large part on classical recursion theory. However, in its concern for effectiveness, this work represents an extension of, rather than a part of, classical descriptive set theory. (The work of Hrbacek and Simpson also relies heavily on notions of effectiveness and thus may be regarded as essentially "unclassical".) The present paper initiates the study of an analog to the theory of index sets for recursively enumerable sets in the context of sets of indices for open sets. In ?0, we describe the context in more detail. In ?1, we establish classical analogs for the Rice and Rice-Shapiro theorems, providing normal forms for both clopen and open index sets. This discussion involves no notions of effectiveness and uses no techniques that are less than 50 years old. In ?2 we show that the same arguments can be used to provide an effective version of the same results based on Moschovakis's discussion [6] of semirecursive open sets in effectively presented Polish spaces. The problems of providing normal forms for Borel index sets of greater complexity and of classifying some natural index sets will be treated in [5].