A Study in Grzegorczyk Point-Free Topology Part I: Separation and Grzegorczyk Structures

Abstract

Abstract.This is the first, out of two papers, devoted to Andrzej Grzegorczyk's pointfree system of topology from Grzegorczyk (Synthese 12(2-3): [228][229][230][231][232][233][234][235] 1960. https://doi. org/10.1007/BF00485101). His system was one of the very first fully fledged axiomatizations of topology based on the notions of region, parthood and separation (the dual notion of connection). Its peculiar and interesting feature is the definition of point, whose intention is to grasp our geometrical intuitions of points as systems of shrinking regions of space. In this part we analyze (quasi-)separation structures and Grzegorczyk structures, and establish their properties which will be useful in the sequel. We prove that in the class of Urysohn spaces with countable chain condition, to every topologically interpreted representative of a point in the sense of Grzegorczyk's corresponds exactly one point of a space. We also demonstrate that Tychonoff first-countable spaces give rise to complete Grzegorczyk structures. The results established below will be used in the second part devoted to points and topological spaces.

A comparison of two systems of point-free topology

“…We define Sep := qSep + (S4) + (S5). As it was observed in [3], in all structures from Sep, by (S3) and (S5), the relation is definable by )(, i.e., we have:…”

“…All quasi-connection structures that satisfy (C4) and (C5) are called connection structures (but see Remark 1.1 in [3]). «Standard» quasi-separation (resp.…”

“…separation) structures are obtained from topological spaces by means of the following method. For any topological space T = S, O , the pair rO + , ⊆ is a mereological structure (see for example Lemma 2.9 in [3]). The separation relation…”

“…

This is a spin-off paper to [3,4] in which we carried out an extensive analysis of Andrzej Grzegorczyk's point-free topology from [5]. In [1] Loredana Biacino and Giangiacomo Gerla presented an axiomatization which was inspired by the Grzegorczyk's system, and which is its variation.

…”

“…The symbols 'qSep', 'Sep' and 'G' (see further) are used in[3,4], but there their meaning is different, since in the aforementioned papers we take into account structures which are not complete. Thus the reader who is familiar with the two papers is asked to bear in mind that all structures analyzed in this paper are complete.…”