The main result is the following. Let f:X→Y be a continuous mapping of completely Baire space X onto hereditary weakly Preiss-Simon regular Y such that image every open subset resolvable set in Y. Then Baire. classical Hurewicz theorem about closed embedding rational numbers into metrizable spaces generalized to spaces.

It is proved, under Martin’s Axiom, that all (ω1, ω1) gaps in P(N) / fin are indestructible in any forcing extension by a separable measure algebra. This naturally leads to a new type of gap, a summable gap. The results of these investigations have applications in Descriptive Set Theory. For example, it is shown that under Martin’s Axiom the Baire categoricity of all ∆3 non-∆ 1 3-complete sets ...

Perhaps the most strikingly counterintuitive theorem in mathematics is the "Banach-Tarski paradox": A ball in R3 can be decomposed into finitely many pieces which can be rearranged by rigid motions and reassembled to form two balls of the same size as the original. The Axiom of Choice is used to construct the decomposition; the "paradox" is resolved by noting that the pieces cannot be Lebesgue ...

A classical theorem of Kuratowski says that every Baire one function on a Gδ subspace of a Polish (= separable completely metrizable) space X can be extended to a Baire one function on X. Kechris and Louveau introduced a finer gradation of Baire one functions into small Baire classes. A Baire one function f is assigned into a class in this heirarchy depending on its oscillation index β(f). We p...

Perhaps the earliest results about pointwise compact sets of Baire class-1 functions are the two selection theorems of E. Helly found in most of the standard texts on real variable (see, e.g., [Lo], [N]). These two theorems are really theorems about a particular example of a compact set of Baire class-1 functions known today as Helly space, the space of all nondecreasing functions from the unit...

In his thesis Baire defined functions of Baire class 1. A function f is of Baire class 1 if it is the pointwise limit of a sequence of continuous functions. Baire proves the following theorem. A function f is not of class 1 if and only if there exists a closed nonempty set F such that the restriction of f to F has no point of continuity. We prove the automaton version of this theorem. An ω-rati...

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...

We discuss the connection of additive set functions to dual function spaces by means of integrals. Isometrical isomorphisms are presented between: a) bounded contents and the dual of uniformly approximable functions, b) charges on Baire algebras and the duals of bounded, continuous functions over A-topological spaces, c) Hewitt measures on Baire σ-algebras and the duals of continuous functions ...

The following discusses a generalization of Baire category in a continuous set. The objective is to provide a meaningful classification of subsets of a continuous set as “large” or “small” sets in linearly ordered continuous sets. In particular, for cardinal number κ, the continuous ordered set 2∗ a subset of the set of dyadic sequences of length κ is discussed. We establish that this space, an...

The purposes of this note are to complement the results of [4] by citing examples of spectral operators (in the sense of that paper) on non-normable locally convex spaces and to announce a result on the structure of spectral operators which indicates that these examples exhibit rather typical behavior for spectral operators on a large class of locally convex spaces of interest to analysts. We r...