In ZF (i.e., Zermelo-Fraenkel set theory without the Axiom of Choice) the following statements are shown to be equivalent: (1) The axiom of dependent choice. (2) Products of compact Hausdorff spaces are Baire. (3) Products of pseudocompact spaces are Baire. (4) Products of countably compact, regular spaces are Baire. (5) Products of regular-closed spaces are Baire. (6) Products of Čech-complete...

Abstract We introduce the notions of ? -Baire property and -semiopen set using sets Lebesgue measure zero. For a family A subsets real line, we define ( ? )-property analogously as it was done in category case for )-property. The main result is that all line having has iff situated between Euclidean topology sets.

in proposition 2.6 in (g‎. ‎gruenhage‎, ‎a‎. ‎lutzer‎, ‎baire and volterra spaces‎, ‎textit{proc‎. ‎amer‎. ‎math‎. ‎soc.} {128} (2000)‎, ‎no‎. ‎10‎, ‎3115--3124) a condition that‎ ‎every point of $d$ is $g_delta$ in $x$ was overlooked‎. ‎so we‎ ‎proved some conditions by which a baire space is equivalent to a‎ ‎volterra space‎. ‎in this note we show that if $x$ is a‎ ‎monotonically normal $t_1$...

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 ...

For any countable group satisfying the \weak Rohlin property", and for each dynamical property, the set of-actions with that property is either residual or meager. The class of groups with the weak Rohlin property includes each lattice Z d ; indeed, all countable discrete amenable groups. For an arbitrary countable group, let A be the set of-actions on the unit circle Y. We establish an Equival...

We study some set-theoretic properties of Schmidt’s σ-ideal on R, emphasizing its analogies and dissimilarities with both the classical σ-ideals on R of Lebesgue measure zero sets and of Baire first category sets. We highlight the strict analogy between Schmidt’s ideal on R and Mycielski’s ideal on 2N.

The Baire theory of category, which classifies sets into two distinct categories, is an important topic in the study of metric spaces. Many results in topology arise from category theory; in particular, the Baire categories are related to a topological property. Because the Baire Category Theorem involves nowhere dense sets in a complete metric space, this paper first develops the concepts of n...

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...