Bitopological and topological ordered k-spaces

Domain theory, in theoretical computer science, needs to be able to handle function spaces easily. It also requires asymmetric spaces, and these are necessarily not T1. At the same time, techniques used with the higher separation axioms are useful there (see [Topology Appl. 199 (2002) 241]). In order to handle all these requirements, we develop a theory of k-bispaces using bitopological spaces, which results in a Cartesian closed category. The other well-known way to combine asymmetry and separation is ordered topological spaces [Nachbin, Topology and Order, Van Nostrand, 1965]; we define the category of ordered k-spaces, which is isomorphic to that found among bitopological spaces. © 2004 Elsevier B.V. All rights reserved

Foreword

Digitalitzat per Artypla

Ordered Products of Topological Groups

The topology most often used on a totally ordered group (G, \u3c) is the interval topology. There are usually many ways to totally order G x G (e.g., the lexicographic order) but the interval topology induced by such a total order is rarely used since the product topology has obvious advantages. Let ℝ(+) denote the real line with its usual order and Q(+) the subgroup of rational numbers. There is an order on Q x Q whose associated interval topology is the product topology, but no such order on ℝ x ℝ can be found. In this paper we characterize those pairs G, H of totally ordered groups such that there is a total order on G x H for which the interval topology is the product topology

Auxiliary relations and sandwich theorems

A well-known topological theorem due to Katv etov states: Suppose $(X,tau)$ is a normal topological space, and let $f:Xto[0,1]$ be upper semicontinuous, $g:Xto[0,1]$ be lower semicontinuous, and $fleq g$. Then there is a continuous $h:Xto[0,1]$ such that $fleq hleq g$. We show a version of this theorem for many posets with auxiliary relations. In particular, if $P$ is a Scott domain and $f,g:Pto[0,1]$ are such that $fleq g$, and $f$ is lower continuous and $g$ Scott continuous, then for some $h$, $fleq hleq g$ and $h$ is both Scott and lower continuous. As a result, each Scott continuous function from $P$ to $[0,1]$, is the sup of the functions below it which are both Scott and lower continuous

The Space of Minimal Prime Ideals of C(x) Need not be Basically Disconnected

Problems posed twenty and twenty-five years ago by M. Henriksen and M. Jerison are solved by showing that the space of minimal prime ideals of the ring C(X) of continuous real-valued functions on a compact (Hausdorff) space need not be basically disconnected-or even an F-space

Topologies and Cotopologies Generated by Sets of Functions

Let L be either [0, 1] or {0, 1} with the usual order. We study topologies on a set X for which the cozero-sets of certain subfamilies H of Lx form a base, and the properties imposed on such topologies by hypothesizing various order-theoretic conditions on H. We thereby obtain useful generalizations of extremely disconnected spaces, basically disconnected spaces, and F-spaces. In particular we use these tools to study the space of minimal prime ideals of certain commutative rings

06341 Abstracts Collection -- Computational Structures for Modelling Space, Time and Causality

From 20.08.06 to 25.08.06, the Dagstuhl Seminar 06341 ``Computational Structures for Modelling Space, Time and Causality\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

04351 Summary -- Spatial Representation: Discrete vs. Continuous Computational Models

Topological notions and methods are used in various areas of the physical sciences and engineering, and therefore computer processing of topological data is important. Separate from this, but closely related, are computer science uses of topology: applications to programming language semantics and computing with exact real numbers are important examples. The seminar concentrated on an important approach, which is basic to all these applications, i.e. spatial representation

04351 Abstracts Collection -- Spatial Representation: Discrete vs. Continuous Computational Models

From 22.08.04 to 27.08.04, the Dagstuhl Seminar 04351 ``Spatial Representation: Discrete vs. Continuous Computational Models\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Partial metrizability in value quantales

[EN] Partial metrics are metrics except that the distance from a point to itself need not be 0. These are useful in modelling partially defined information, which often appears in computer science. We generalize this notion to study “partial metrics” whose values lie in a value quantale which may be other than the reals. Then each topology arises from such a generalized metric, and for each continuous poset, there is such a generalized metric whose topology is the Scott topology, and whose dual topology is the lower topology. These are both corollaries to our result that a bitopological space is pairwise completely regular if and only if there is such a generalized metric whose topology is the first topology, and whose dual topology is the second.This author wishes to acknowledge support for this research from the EPSRC of the United Kingdom (grant GR/S07117/01), and from the City University of New York (PSCCUNY grant 64472-00 33).Kopperman, RD.; Matthews, S.; Pajoohesh, H. (2004). Partial metrizability in value quantales. Applied General Topology. 5(1):115-127. https://doi.org/10.4995/agt.2004.2000SWORD1151275