Generalized closed sets and weak separation axioms
by
Julian Dontchev
University of Helsinki
Coauthors: Maximilian Ganster, Francisco G. Arenas, Maria Luz Puertas

Separation axioms stand among the most common and to a certain extent the most important and interesting concepts in Topology. The definitions of most (if not all) weak separation axioms are deceptively simple. However, the structure and the properties of those spaces are not always that easy to comprehend. Most weak separation axioms are defined in terms of generalized closed sets. In Digital Topology, several spaces that fail to be T₁ are important in the study of the geometric and topological properties of digital images. Although the digital line (the major building block of the digital n-space) is neither a T₁-space nor an R₀-space, it satisfies a couple of separation axioms which are a bit weaker than T₁ and R₀, that is, the digital line is both a T_\frac34-space (hence a semi-T₁-space) and a semi-R₀-space. This inclines to indicate that further knowledge of the behavior of topological spaces satisfying these two weak separation axioms (and some related ones) is required. This is indeed the intention of this talk. We try to unify the separation axioms between T₀ and completely Hausdorff by introducing the concept of T_{\kappa, \xi}-spaces. We call a topological space (X, \tau) a T_{\kappa, \xi}-space if every compact subset of X with cardinality <= \kappa is \xi-closed, where \xi is a given closure operator. With different settings on \kappa and \xi we derive most of the well-known separation properties `in the semi-closed interval [T₀, T₃)'. We are going to consider not only Kuratowski closure operators but more general closure operators, such as the \lambda-closure operator for example.

Date received: May 25, 1998

Copyright © 1998 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cabc-19.