Generalized continuous closure spaces I

Meet preserving closure operations

authored by

Marcel Erné

Abstract

We study general notions of convergence and continuity in arbitrary spaces or ordered sets, extending considerably topological concepts in domain theory like those of Scott convergence, alias lower (lim-inf) convergence, and Scott topology. It turns out that the convergence-theoretical properties of being localized, a limit relation, pretopological, or topological, respectively, all correspond to important properties of the underlying ordered sets that reduce to (meet) continuity and similar properties in the classical situation. Basic tools are the cut closure operators and diverse order-theoretical or topological variants of them. We characterize the generalized Scott convergence spaces abstractly as so-called core determined convergence spaces. This unifying concept provides simplifications and new insights into various areas of order theory, topology and theoretical computer science. In particular, some intimate connections between convergence properties, meet preservation by certain closure operations, and the continuity of meet operations are established.

Organisation(s)

Institute of Algebra, Number Theory and Discrete Mathematics

Type

Article

Journal

Topology and its applications

Volume

273

ISSN

0166-8641

Publication date

15.03.2020

Publication status

Published

Peer reviewed

Yes

ASJC Scopus subject areas

Geometry and Topology

Electronic version(s)

https://doi.org/10.1016/j.topol.2019.106981 (Access: Open)