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A Characterization Oof Discrete Topological Spaces as Convexity Spaces by Stanisław Kryński is a Mathematics article available to read on EtoBox.

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## CONVEXITY SPACES 1. Preliminaries 1.1. Axiomatic) convexity Let X be any fixed nonempty set and P{X) -the family of all its subsets. Definition 1 [2j. A convexity\*\* in X is a subfamily £ of-^(X) such that (i) Xe$ (ii) njfeS for every subfamily Jt of 5 . The pair (X,5) is oalled a convexity space, and elements of 5 -convex sets. The classioal convexity in linear spaces, as well as various generalizations of it that oan be met in the literature, satisfy (1) and (2), therefore are convexities in the sense of Definition 1. Many specifio notions of convexity theory defined and discussed in other contexts oan be analyzed in this framework) e.g. convex hull, extreme points, quasiconvex functions e.t.c. The present paper is devoted to semispaoes of a convexity space. ' This oonoept is known in other fields of mathematics under various names, e.g. a "Moore family" (see G. Birkhoff, "Lattioe Theory"), or a "closure system" (P. Cohn, "Universal Algebra")j also cf. a "cyrtology" int

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Author

Stanisław Kryński

Published

1989

Language

EN

Field

Mathematics (Physical Sciences)