In Domain Theory quasicontinuous domains pop up from time to time generalizing slightly the powerful notion of a continuous domain. It is the aim of this paper to show that quasicontinuous domains occur in a natural way in relation to the powerdomains of finitely generated and compact saturated subsets. Properties of quasicontinuous domains seem to be best understood from that point of view. Th...

Xavier Cabré Abstra t. We consider a class of second order linear elliptic operators intrinsically defined on Riemannian manifolds, and which correspond to nondivergent operators in Euclidean space. Under the assumption that the sectional curvature is nonnegative, we prove a global Krylov-Safonov Harnack inequality and, as a consequence, a Liouville theorem for solutions of such equations. From...

Let H(X, Y ) ( C(X, Y ) ) represent the family of holomorphic (continuous) maps from a complex (topological) space X to a complex (topological) space Y , and let Y + = Y ∪{∞} be the Alexandroff one–point compactification of Y if Y is not compact, Y + = Y if Y is compact. We say that F ⊂ H(X, Y ) is uniformly normal if {f ◦ φ : f ∈ F , φ ∈ H(M,X)} is relatively compact in C(M,Y +) (with the comp...

This paper is devoted to a study of infinite horizon optimal control problems with time discounting and averaging criteria in discrete time. It known that these are related certain infinite-dimensional linear programming problems, but compactness the state constraint common assumption imposed analysis LP problems. In this paper, we consider an unbounded use Alexandroff compactification carry ou...

The Hausdorff–Alexandroff Theorem states that any compact metric space is the continuous image of Cantor’s ternary set C. It is well known that there are compact Hausdorff spaces of cardinality equal to that of C that are not continuous images of Cantor’s ternary set. On the other hand, every compact countably infinite Hausdorff space is a continuous image of C. Here we present a compact counta...

In this paper we explore the topological semantics for conditional logic that arises from the Alexandroff equivalence between preorders and topological spaces. This clarifies the relation between the standard order semantics and premise semantics for conditionals. As an application we provide a construction of relative similarity orders between possible worlds from topologies of relevant propos...

A non-empty subset of a topological space is irreducible if whenever it is covered by the union of two closed sets, then already it is covered by one of them. Irreducible sets occur in proliferation: (1) every singleton set is irreducible, (2) directed subsets (which of fundamental status in domain theory) of a poset are exactly its Alexandroff irreducible sets, (3) directed subsets (with respe...