### A dense subsemigroup of S(R) generated by two elements

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A closed subset of the real line which is right porous but is not $\sigma $-left-porous is constructed.

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.

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 ${\alpha}_{1.5}$ 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 ${C}_{p}\left(X\right)$ of continuous real-valued functions on X with the topology...

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,...

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...

We construct a set B and homeomorphism f where f and ${f}^{-1}$ have property N such that the symmetric difference between the sets of density points and of f-density points of B is uncountable.

We formulate a Covering Property Axiom $CP{A}_{cube}$, 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 $\u27e8f\u2099\in {\mathbb{R}}^{\mathbb{R}}{\u27e9}_{n<\omega}$ of Borel functions there are sequences:...

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...