Open Subsets of LF-spaces

On the intermediate logic of open subsets of metric spaces

In this paper we study the intermediate logic MLO(X ) of open subsets of a metric space X . This logic is closely related to Medvedev’s logic of finite problemsML. We prove several facts about this logic: its inclusion in ML, impossibility of its finite axiomatization and indistinguishability from ML within some large class of propositional formulas.

Grey Subsets of Polish Spaces

We develop the basics of an analogue of descriptive set theory for functions on a Polish space X. We use this to define a version of the small index property in the context of Polish topometric groups, and show that Polish topometric groups with ample generics have this property. We also extend classical theorems of Effros and Hausdorff to the topometric context.

Analytic Subsets of Hilbert Spaces †

We show that every complete metric space is homeomorphic to the locus of zeros of an entire analytic map from a complex Hilbert space to a complex Banach space. As a corollary, every separable complete metric space is homeomorphic to the locus of zeros of an entire analytic map between two complex Hilbert spaces. §1. Douady had observed [8] that every compact metric space is homeomorphic to the...

Topologies on Spaces of Subsets

The Banach-Steinhaus theorem for (LF)-spaces in constructive analysis

The space D(R) of test functions (infinitely differentiable functions with compact support) is an important example of a non-metrizable (LF)-space, and the domain of distributions (generalized functions). E. Bishop suggested in [1, Appendix A] and [2, Chapter 7, Notes] that the completeness of D(R) and the weak completeness of its dual space would not hold in Bishop’s constructive mathematics. ...