A dense subsemigroup of S(R) generated by two elements
A Synopsis of Elementary Results in Pure Mathematics [Book]
A -porous set need not be -bilaterally porous
A closed subset of the real line which is right porous but is not -left-porous is constructed.
Alternative definitions of -density topology
Analytic Baire spaces
We generalize to the non-separable context a theorem of Levi characterizing Baire analytic spaces. This allows us to prove a joint-continuity result for non-separable normed groups, previously known only in the separable context.
Application of covering sets
Arhangel'skiĭ sheaf amalgamations in topological groups
We consider amalgamation properties of convergent sequences in topological groups and topological vector spaces. The main result of this paper is that, for arbitrary topological groups, Nyikos’s property is equivalent to Arhangel’skiĭ’s formally stronger property α₁. This result solves a problem of Shakhmatov (2002), and its proof uses a new perturbation argument. We also prove that there is a topological space X such that the space of continuous real-valued functions on X with the topology...
Baire classification of ...-density and ...-approximately continous functions.
Beyond Lebesgue and Baire: generic regular variation
We show that the No Trumps combinatorial property (NT), introduced for the study of the foundations of regular variation by the authors, permits a natural extension of the definition of the class of functions of regular variation, including the measurable/Baire functions to which the classical theory restricts itself. The "generic functions of regular variation" defined here characterize the maximal class of functions to which the three fundamental theorems of regular variation (Uniform Convergence,...
Beyond Lebesgue and Baire II: Bitopology and measure-category duality
We re-examine measure-category duality by a bitopological approach, using both the Euclidean and the density topologies of the line. We give a topological result (on convergence of homeomorphisms to the identity) obtaining as a corollary results on infinitary combinatorics due to Kestelman and to Borwein and Ditor. We hence give a unified proof of the measure and category cases of the Uniform Convergence Theorem for slowly varying functions. We also extend results on very slowly varying functions...
Cluster sets of arbitrary functions defined on plane sets
Construction of an Uncountable Difference between Φ(B) and
We construct a set B and homeomorphism f where f and have property N such that the symmetric difference between the sets of density points and of f-density points of B is uncountable.
Constructive irrational space.
Continuity and theorem of Heine-Cantor.
Convergences for the group of real numbers
Covering Property Axiom and its consequences
We formulate a Covering Property Axiom , which holds in the iterated perfect set model, and show that it implies easily the following facts. (a) For every S ⊂ ℝ of cardinality continuum there exists a uniformly continuous function g: ℝ → ℝ with g[S] = [0,1]. (b) If S ⊂ ℝ is either perfectly meager or universally null then S has cardinality less than . (c) cof() = ω₁ < , i.e., the cofinality of the measure ideal is ω₁. (d) For every uniformly bounded sequence of Borel functions there are sequences:...
Directional qualitative cluster sets
Expansions of the real line by open sets: o-minimality and open cores
The open core of a structure ℜ := (ℝ,<,...) is defined to be the reduct (in the sense of definability) of ℜ generated by all of its definable open sets. If the open core of ℜ is o-minimal, then the topological closure of any definable set has finitely many connected components. We show that if every definable subset of ℝ is finite or uncountable, or if ℜ defines addition and multiplication and every definable open subset of ℝ has finitely many connected components, then the open core of ℜ is...
Fundamental theorems of analysis in formally real fields.