Of these definitions, the first and third may make some sense, since the definition of a filter base describes the family C of sets in the “generalized Cantor intersection theorem” except for dropping the requirement that they be closed (which would not make sense in a general set X anyway). The definition of a filter is motivated in two ways. The first is that, since one is only interested in ...

A Tychonoff space X is called RG if the embedding of C(X) → C(Xδ) is an epimorphism of rings. Compact RG spaces are known and easily described. We study the pseudocompact RG spaces. These must be scattered of finite Cantor Bendixon degree but need not be locally compact. However, under strong hypotheses, (countable compactness, or small cardinality) these spaces must, indeed, be compact. The ma...

We prove that a ‘small’ extension of a minimal AF equivalence relation on a Cantor set is orbit equivalent to the AF relation. By a ‘small’ extension we mean an equivalence relation generated by the minimal AF equivalence relation and another AF equivalence relation which is defined on a closed thin subset. The result we obtain is a generalization of the main theorem in [GMPS2].

We study in this paper ordered finite measure algebras from the point of view of Fräıssé and Ramsey theory. We also propose an open problem, which is a homogeneous version of the Dual Ramsey Theorem of GrahamRothschild, and derive consequences of a positive answer to the study of the topological dynamics of the automorphism group of a standard probability space and also the group of measure pre...

In this paper we consider families of compact convex sets in R such that any subfamily of size at most d has a nonempty intersection. We prove some analogues of the central point theorem and Tverberg’s theorem for such families.

The mathematical concept of infinity, in the sense Cantor, is rather far from applied mathematics and statistics. These fields can be linked. We comment on properties infinite numbers relate them to some operations with random variables. existence statistical parametric models studied terms cardinal numbers. Some probabilistic interpretations Gödel’s theorem, Turing’s halting problem, Banach-Ta...

We introduce a notion of weak intersection number of a collection of sets, modifying the notion of intersection number due to J.L. Kelley, and obtain an analogue of Kelley’s characterization of Boolean algebras which support a finitely additive strictly positive measure. We also consider graph-theoretic reformulations of the notions of intersection number and weak intersection number. Kelley [4...

In analogy to the Eisenfeld-Lakshmikanthammeasure of nonconvexity and theHausdorffmeasure of noncompactness, we introduce two mutually equivalent measures of noncircularity for Banach spaces satisfying a Cantor type property, and apply them to establish a fixed point theorem of Darbo type for multifunctions. Namely, we prove that every multifunction with closed values, defined on a closed set a...

In this note we consider dynamical systems (X, G) on a Cantor set X satisfying some mild technical conditions. The considered class includes, in particular, minimal and transitive aperiodic systems. We prove that two such systems (X1, G1) and (X2, G2) are orbit equivalent if and only if their full groups are isomorphic as abstract groups. This result is a topological version of the well-known D...