RECURSIVE BAIRE CLASSIFICATION AND SPEEDABLE FUNCTIONS

Using recursive variants of Baire notions of nowhere dense and meagre sets we study the topological size of speedable and infinitely often speedable functions in a machine-independent framework. We show that the set of speedable functions is not “small” whereas the set of infinitely often speedable functions is “large”. In this way we offer partial answers to a question in [4].     

Abstract functions with continuous differences and Namioka spaces

Let be a semigroup and a topological space. Let be an Abelian topological group. The right differences of a function are defined by for . Let be continuous at the identity of for all in a neighbourhood of . We give conditions on or range under which is continuous for any topological space . We also seek conditions on under which we conclude that is continuous at for arbitrary . This led us to introduce new classes of semigroups containing all complete metric and locally countably compact quasitopological groups. In this paper we study these classes and explore their relation with Namioka spaces.

     

On translations of subsets of the real line

In this paper we discuss various questions connected with translations of subsets of the real line. Most of these questions originate from W. Sierpinski. We discuss the number of translations a single subset of the reals may have. Later we discuss almost invariant subsets of Abelian groups.

     

Sets of uniqueness for spherically convergent multiple trigonometric series

A subset of the -dimensional torus is called a set of uniqueness, or -set, if every multiple trigonometric series spherically converging to outside vanishes identically. We show that all countable sets are -sets and also that sets are -sets for every . In particular, , where is the Cantor set, is an set and hence a -set. We will say that is a -set if every multiple trigonometric series spherically Abel summable to outside and having certain growth restrictions on its coefficients vanishes identically. The above-mentioned results hold also for sets. In addition, every -set has measure , and a countable union of closed -sets is a -set.

     

A Moore space with a -base which cannot be densely embedded in any Moore space with the Baire property

The author answers a question raised in the literature about twenty five years ago and raised again more recently in Open Problems in Topology, by G. M. Reed, concerning the conjecture that every Moore space with a -discrete -base can be densely embedded in a Moore space having the Baire property. Even though closely related results have made this conjecture seem likely to be true, the author shows that, surprisingly, the conjecture is false.

     

Transfinite sequences of continuous and Baire class 1 functions

The set of continuous or Baire class 1 functions defined on a metric space is endowed with the natural pointwise partial order. We investigate how the possible lengths of well-ordered monotone sequences (with respect to this order) depend on the space .

     

Universally meager sets

We study category counterparts of the notion of a universal measure zero set of reals.

We say that a set is universally meager if every Borel isomorphic image of is meager in . We give various equivalent definitions emphasizing analogies with the universally null sets of reals.

In particular, two problems emerging from an earlier work of Grzegorek are solved.

     

Some Generic Results on Nonattaining Functionals

We prove that (a) in a reflexive space, for any linearly bounded but unbounded closed convex subset the nonsupport functionals are a dense G subset of the polar set, and (b) any nonsemicoercive proper convex lsc [weak^*-lsc] function in a [dual] Banach space has a generic [dense G ] set of L -perturbations which do not attain their infimum. We also characterize the proper convex functions that have inf-nonattaining L -perturbations. This results also in a criterion for reflexivity.     

Completeness properties of the pseudocompact-open topology on <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Italic">X</Emphasis>)

This is a study of the completeness properties of the space C _ps (X) of continuous real-valued functions on a Tychonoff space X, where the function space has the pseudocompact-open topology. The properties range from complete metrizability to the Baire space property. Dedicated to Professor Robert A. McCoy     

On free semigroups of automaton transformations

It is established that the subset of freek-generated subsemigroups of the semigroup of all automaton transformations over a finite alphabet is a second category set (in the sense of the Baire category approach) in the set of allk-generated subsemigroups. A continuum series of pairs of automaton transformations each of which generates a free semigroup of rank two is indicated. A criterion is established for this semigroup to be a finite-automaton group. Translated fromMatematicheskie Zametki, Vol. 63, No. 2, pp. 248–259, February, 1998. The author wishes to express his deep gratitude to Professor V. I. Sushchans'kii for permanent help and attention to the research. This research was partially supported by the ISSEP under grant No. GSU 051341.