In this paper, the concept of somewhat-connected space will be introduced and characterized. Its connection with the other well-known concepts such as the classical connectedness, the $omega_theta$-connectedness, and the $omega$-connectedness will be determined. Moreover, the concept of textit{somewhat}-continuous function from an arbitrary topological space into the product space will be chara...

A theorem on the existence of separable supports of σ-finite Borel measures given on metric spaces with small topological weights is applied to constructions of certain analogues of special subsets of the real line. A well-known result from topological measure theory states that every σ-finite Borel measure on a metric space whose weight is not real-valued measurable possesses a separable suppo...

We develop further the topological theory of regular variation of [BOst13]. There we established the uniform convergence theorem (UCT) in the setting of topological dynamics (i.e. with a group T acting on a homogenous space X), thereby unifying and extending the multivariate regular variation literature. Here, working with realtime topological ‡ows on homogeneous spaces, we identify an index of...

In a previous paper the authors developed an operator-algebraic approach to Lewis Bowen’s sofic measure entropy that yields invariants for actions of countable sofic groups by homeomorphisms on a compact metrizable space and by measure-preserving transformations on a standard probability space. We show here that these measure and topological entropy invariants both coincide with their classical...

With a new proof approach we prove in more general setting the classical convergence theorem that almost everywhere of measurable functions on finite measure space implies measure. Specifically, generalize for case where codomain is separable metric and limiting map constant an arbitrary topological space.

Let T be a compact Hausdorff topological space and let M denote an n–dimensional subspace of the space C(T ), the space of real–valued continuous functions on T and let the space be equipped with the uniform norm. Zukhovitskii [7] attributes the Basic Theorem to E.Ya.Remez and gives a proof by duality. He also gives a proof due to Shnirel’man, which uses Helly’s Theorem, now the paper obtains a...

The regular open subsets of a topological space form Boolean algebra, where the join two sets is interior closure their union. A content finitely additive measure on this or one its subalgebras. We develop theory integration for such contents. then explain relationship between contents, residual charges, and Borel measures. show that can be represented by normal measure, augmented with liminal ...

A unitary operator U acting on the weakly compact unit ball B of a Hilbert space H gives a topological dynamical system (B, V). We prove in this note that for every ergodic u-invariant measure m, the dynamical system (B, U, m) is conjugated to a strictly ergodic group translation. Especially, the topological entropy of (B, U) is zero. As an application, it follows that any flow on the unit ball...