These are notes of the author’s talk on various types of spectra associated naturally with the eventually domination ordering on the Baire space ωω , given at the General Topology Symposium at Kobe University in December 2002. The report comes in two parts: in the first half, we present an outline of the lecture, giving ideas of some of the arguments without going too deeply into details. The s...

A connection between the separability and the countable chain condition of spaces with L-property (a topological space X has L-property if for every topological space Y , separately continuous function f : X ×Y →R and open set I ⊆R, the set f −1(I) is an Fσ-set) is studied. We show that every completely regular Baire space with the L-property and the countable chain condition is separable and c...

I give an overview/summary of the techniques used, and the results derived, in my 1984 PhD dissertation, Reducibility and Determinateness on the Baire Space. In particular, I focus on the calculation of the order type (and structure) of the collection of degrees of Borel sets.

Suppose that X is a Polish space and E is a countable Borel equivalence relation on X. We show that if there is a Borel assignment of means to the equivalence classes of E, then E is smooth. We also show that if there is a Baire measurable assignment of means to the equivalence classes of E, then E is generically smooth.

We prove the consistency of a strong uniformization principle for a subset of the Baire space of cardinality א1. As a consequence we get the consistency of a related group-theoretic principle. Date: January 18, 2015. 2010 Mathematics Subject Classification. Primary, Secondary.

We introduce the concept of Baire embeddings and we classify them up to C conjugacies. We show that two such embeddings are Cequivalent if and only if they have exponentially equivalent geometries. Next, we introduce the class of IFS-like Baire embeddings and we also show that two Hölder equivalent IFS-like Baire embeddings are C conjugate if and only if their scaling functions are the same. In...

For a compact metric space ∆ we consider the compact space S(∆) of shift-invariant subsystems of the full shift ∆, endowed with the Hausdorf metric.. We show that when ∆ is perfect the space S(∆) and the homeomorphism group Homeo(K) of the Cantor set (with the topology of uniform convergence) are topologically generically equivalent in the sense of Glasner and King [7], ie a dynamical property ...

According to a result of Kočinac and Scheepers, the Hurewicz covering property is equivalent to a somewhat simpler selection property: For each sequence of large open covers of the space one can choose finitely many elements from each cover to obtain a groupable cover of the space. We simplify the characterization further by omitting the need to consider sequences of covers: A set of reals X sa...

In this note we show that ω1-Universally Baire selfjustifying systems are fully Universally Baire under the Weak Stationary Reflection Principle for Pairs. This involves analyzing the notion of a weakly captured set of reals, a weakening of the Universal Baire property.