A non-connected, Hausdorff space with a countable network has a connected Hausdorff-subtopology iff the space is not-H-closed. This result answers two questions posed by Tkačenko, Tkachuk, Uspenskij, and Wilson [Comment. Math. Univ. Carolinae 37 (1996), 825–841]. A non-H-closed, Hausdorff space with countable π-weight and no connected, Hausdorff subtopology is provided.

We introduce and analyze a new cardinal characteristic of the continuum, splitting number reals, denoted s(R). This is connected to Efimov's problem, which asks whether every infinite compact Hausdorff space must contain either non-trivial convergent sequence, or else copy βN.

In this paper the notion of s-irrelevance with respect to upper and lower conditional probabilities assigned by Hausdorff outer and inner measures is proved to be a sufficient condition for strong independence introduced for credal sets. An example is given to show that the converse is not true. Moreover the definition of sconditional irrelevance is given and a generalized factorization propert...

Let {xn}n∈N be a sequence of [0, 1]d, {λn}n∈N a sequence of positive real numbers converging to 0, and δ > 1. The classical ubiquity results are concerned with the computation of the Hausdorff dimension of limsup-sets of the form S(δ) = T N∈N S n≥N B(xn, λ δ n). Let μ be a positive Borel measure on [0, 1]d, ρ ∈ (0, 1] and α > 0. Consider the finer limsup-set

In this paper we prove that, given s ≥ 0, and a Borel non zero measure μ in Rm, if for μ-almost every x ∈ Rm the limit lim ε→0 ∫ |x−y|>ε x −y |x −y|s+1 dμ(y) exists and 0 < lim supr→0 μ(B(x, r))/r s < ∞, then s in an integer. In particular, if E ⊂ Rm is a set with positive and bounded s-dimensional Hausdorff measure Hs and for Hs-almost every x ∈ E the limit

In this thesis, we introduce a finitary logic which is sound and complete with respect to de Vries algebras, and hence by de Vries duality, this logic can be regarded as logic of compact Hausdorff spaces. In order to achieve this, we first introduce a system S which is sound and complete with respect to a wider class of algebras. We will also define Π2-rules and establish a connection between Π...

Previously, the authors used insights of Robinson's nonstandard analysis as a powerful tool to extend and simplify construction some compactifications regular spaces. They now show that any Hausdorff compactification is obtainable with their method.