Related papers: Probabilistic Powerdomains and Quasi-Continuous Do…

Let $S$ be a Scott set, or even an $\omega$-model of $\mathsf{WWKL}$. Then for each $A\in S$, either there is $X \in S$ that is weakly 2-random relative to $A$, or there is $X\in S$ that is 1-generic relative to $A$. It follows that if…

We consider properties and applications of a compact, Hausdorff topology called the "ultrafilter topology" defined on an {\sl arbitrary spectral space} and we observe that this topology coincides with the constructible topology. If $K$ is a…

We study the relationship between generalizations of $P$-spaces and Volterra (weakly Volterra) spaces, that is, spaces where every two dense $G_\delta$ have dense (non-empty) intersection. In particular, we prove that every dense and every…

For a quantale ${\sf{V}}$, the category $\sf V$-${\bf Top}$ of ${\sf{V}}$-valued topological spaces may be introduced as a full subcategory of those ${\sf{V}}$-valued closure spaces whose closure operation preserves finite joins. In…

We prove that for each Polish space X, the space C(X) of continuous real-valued functions on X satisfies a strong version of the Pytkeev property, if endowed with the compact-open topology. (This shows that whereas it need not be…

There exists a function f: N -> N such that for every positive integer d, every quasi-finite field K and every projective hypersurface X of degree d and dimension at least f(d), the set X(K) is non-empty. This is a special case of a more…

We study the semicontinuity of automorphism groups for perturbations of domains in complex space or in complex manifolds. We provide a new approach to the study of such results for domains having minimal boundary smoothness. The emphasis in…

We call a space $X$ {\it weakly linearly Lindel\"of} if for any family $\mathcal{U}$ of non-empty open subsets of $X$ of regular uncountable cardinality $\kappa$, there exists a point $x\in X$ such that every neighborhood of $x$ meets…

A map $f:X\to Y$ between topological spaces is called weakly discontinuous if each subspace $A\subset X$ contains an open dense subspace $U\subset A$ such that the restriction $f|U$ is continuous. A bijective map $f:X\to Y$ between…

A classical theorem of P. McMullen describes all valuations on polytopes that are invariant under translations and weakly continuous, i.e., continuous with respect to parallel displacements of the facets of a polytope. While it is typically…

Given a topological space $X$, we study the structure of $\infty$-convex subsets in the space $SC_p(X)$ of scatteredly continuous functions on $X$. Our main result says that for a topological space $X$ with countable strong fan tightness,…

Let $X$ be a compact metric space and $f:X\to X$ a homeomorphism on $X$. We construct a fundamental domain for the set with finite peaks for each cocycle induced by $\phi\in C(X,R)$. In particular we prove that if a partially hyperbolic…

Domain theory has a long history of applications in theoretical computer science and mathematics. In this article, we explore the relation of domain theory to probability theory and stochastic processes. The goal is to establish a theory in…

Quasi-probabilities appear across diverse areas of physics, but their conceptual foundations remain unclear: they are often treated merely as computational tools, and operations like conditioning and Bayes' theorem become ambiguous. We…

The article is devoted to the study of mappings that satisfy the so-called inverse Poletsky inequality. We consider mappings of quasiextremal distance domains, domains with a locally quasiconformal boundary, and domains which are regular in…

We prove that, for every cardinal number $\alpha\geq {\mathfrak c}$, there exists a metrizable space $X$ with $|X|=\alpha$ such that for every pair of quasiorders $\leq_1$, $\leq_2$ on a set $Q$ with $|Q| \leq \alpha$ satisfying the…

We construct a family of Hausdorff spaces such that every finite product of spaces in the family (possibly with repetitions) is CLP-compact, while the product of all spaces in the family is non-CLP-compact. Our example will yield a single…

We consider locally equi-continuous strongly continuous semigroups on locally convex spaces (X,tau). First, we show that if (X,tau) has the property that weak* compact sets of the dual are equi-continuous, then strong continuity of the…

We give a definition of weak geodesics on prox-regular subsets of Riemannian manifolds as continuous curves with some weak regularities. Then obtaining a suitable Lipschitz constant of the projection map, we characterize weak geodesics on a…

We investigate some basic descriptive set theory for countably based completely quasi-metrizable topological spaces, which we refer to as quasi-Polish spaces. These spaces naturally generalize much of the classical descriptive set theory of…