Strong variants of the Operator Speed-up Theorem, Operator Gap Theorem and Compression Theorem are obtained using an effective version of Baire Category Theorem. It is also shown that all complexity classes of recursive predicates have effective measure zero in the space of recursive predicates and, on the other hand, the class of predicates with almost everywhere complexity above an arbitrary ...

We consider a partition of a space X consisting of a meager subset of X and obtain a sufficient condition for the existence of a subfamily of this partition which gives a non-Baire subset of X. The condition is formulated in terms of the theory of J. Morgan [1]. All notions concerning category bases come from Morgan’s monograph (see [1]). We establish the following theorem. Theorem A. Let (X,S)...

We show that every locally finite bipartite Borel graph satisfying a strengthening of Hall’s condition has a Borel perfect matching on some comeager invariant Borel set. We apply this to show that if a group acting by Borel automorphisms on a Polish space has a paradoxical decomposition, then it admits a paradoxical decomposition using pieces having the Baire property. This strengthens a theore...

The paper develops a new approach to the classical bang-bang theorem in linear control theory, based on Baire category. Among all controls which steer the system from the origin to a given point x̄, we consider those which minimize an auxiliary linear functional φ. For all φ in a residual set, we show that the minimizing control is unique and takes values within a set of extreme points.

The Proper Forcing Axiom is a powerful extension of the Baire Category Theorem which has proved highly effective in settling mathematical statements which are independent of ZFC. In contrast to the Continuum Hypothesis, it eliminates a large number of the pathological constructions which can be carried out using additional axioms of set theory. Mathematics Subject Classification (2000). Primary...

We investigate the reverse-mathematical status of a version of the Baire Category Theorem known as BCT. In a 1993 paper Brown and Simpson showed that BCT is provable in RCA0. We now show that BCT is equivalent to RCA0 over RCA ∗ 0 .

We give an elementary proof that there are two topological generators for the full group of every aperiodic hyperfinite probability measure preserving Borel equivalence relation. Our proof explicitly constructs topological generators for the orbit equivalence relation of the irrational rotation of the circle, and then appeals to Dye’s theorem and a Baire category argument to conclude the genera...