We answer in the affirmative [Th. 3 or Corollary 1] the question of L. V. Keldysh [5, p. 648]: can every Borel set X lying in the space of irrational numbers P not Gδ · Fσ and of the second category in itself be mapped onto an arbitrary analytic set Y ⊂ P of the second category in itself by an open map? Note that under a space of the second category in itself Keldysh understood a Baire space. T...

All spaces considered are metric separable and are denoted usually by the letters X, Y , or Z. ω stands for the set of all natural numbers. If a metric separable space is additionally complete, we call it Polish; if it is a continuous image of ω or, equivalently, of a Polish space, it is called Souslin. The main subject of the present paper is the structure of Baire class 1 functions. Recent de...

Let κ be an uncountable cardinal with κ = κ. In this paper we introduce Rκ, a Cauchy-complete real closed field of cardinality 2. We will prove that Rκ shares many features with R which have a key role in real analysis and computable analysis. In particular, we will prove that the Intermediate Value Theorem holds for a non-trivial subclass of continuous functions over Rκ. We propose Rκ as a can...

We introduce two new topologies on ordered sets: the way below topology and weakly way below topology. These are similar in definition to the Scott topology, but are very different if the set is not continuous. The basic properties of these three topologies are compared. We will show that while domain representable spaces must be Baire, this is not the case with the new topologies.

Using the notion of formal ball, we present a few new results in the theory of quasi-metric spaces. With no specific order: every continuous Yoneda-complete quasi-metric space is sober and convergence Choquet-complete hence Baire in its d-Scott topology; for standard quasi-metric spaces, algebraicity is equivalent to having enough center points; on a standard quasi-metric space, every lower sem...

We show that any symmetric, Baire measurable function from the complement of E0 to a finite set is constant on an E0-nonsmooth square. A simultaneous generalization of Galvin’s theorem that Baire measurable colorings admit perfect homogeneous sets and the Kanovei-Zapletal theorem canonizing Borel equivalence relations on E0-nonsmooth sets, this result is proved by relating E0-nonsmooth sets to ...

For any non-totally disconnected Polish space, there is a family of c = 2א0 many Wadge incomparable finite level Borel subsets. If the space is additionally locally compact or locally connected, there is a family of 2 many Wadge incomparable subsets. In this note, we study the Wadge order for Polish spaces which are not totally disconnected. For a fixed Polish space, the Wadge order for the spa...

We define a locally convex space E to have the Josefson-Nissenzweig property (JNP) if identity map (E?, ?(E?, E)) ? ?*(E?, is not sequentially continuous. By classical theorem, every infinite-dimensional Banach has JNP. A characterization of spaces with JNP given. thoroughly study in various function spaces. Among other results we show that for Tychonoff X, Cp(X) iff there weak* null-sequence (...

This paper presents results about the distribution of subsequences which are typical in the sense of Baire. The first main part is concerned with sequences of the type xk = nkα, n1 < n2 < n3 < · · · , mod 1. Improving a result of Šalát we show that, if the quotients qk = nk+1/nk satisfy qk ≥ 1 + ε, then the set of α such that (xk) is uniformly distributed is of first Baire category, i.e. for ge...

Hausdorff’s paradoxical decomposition of a sphere with countably many points removed (the main precursor of the Banach-Tarski paradox) actually produced a partition of this set into three pieces A, B, C such that A is congruent to B (i.e., there is an isometry of the set which sends A to B), B is congruent to C, and A is congruent to B ∪ C. While refining the Banach-Tarski paradox, R. Robinson ...