The conventional definition of a topological metric over a space specifies properties of any measure of “how separated” two points in that space are. Here it is shown how to extend that definition, and in particular the triangle inequality, to concern arbitrary numbers of points. Such a measure of how separated the points within a collection are can be bootstrapped, to measure “how separated” f...

Explicit product formulas are obtained for families of multivariate polynomials which are orthogonal on simplices and on a parabolic biangle in R. These product formulas are shown to give rise to measure algebras which are hypergroups. The article also includes an elementary proof that the Michael topology for the space of compact subsets of a topological space (which is used in the definition ...

A dynamical version of the Bourgain-Fremlin-Talagrand dichotomy shows that the enveloping semigroup of a dynamical system is either very large and contains a topological copy of βN, or it is a “tame” topological space whose topology is determined by the convergence of sequences. In the latter case the dynamical system is called tame. We use the structure theory of minimal dynamical systems to s...

We describe the set of all invariant measures on the spaces of universal countable graphs and on the spaces of universal countable triangles-free graphs. The construction uses the description of the S∞-invariant measure on the space of infinite matrices in terms of measurable function of two variables on some special space. In its turn that space is nothing more than the universal continuous (B...

Introduction. Some years ago Hausdorff, using Hamel bases, showed that any infinite dimensional real Banach space contained a second category linear subspace that was not complete under any equivalent norm [5]. It is shown below that a slight abstraction of his construction leads to the following: (A) a "multiple offender" example (Theorem 3), one that combines several pathological features whi...

This survey article gives a history of the development of the theory con cerning restriction theorems in real analysis The discussion will span the history from Lusin s Theorem through very recent results concerning in tersections of Lipschitz smooth and H older class functions

It is well known that a microperiodic function mapping a topological group into reals, which is continuous at some point is constant. We introduce the notion of a microperiodic multifunction, defined on a topological group with values in a metric space, and study regularity conditions implying an analogous result. We deal with Vietoris and Hausdorff continuity concepts.Stability of microperiodi...