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Studia Logica

, Volume 106, Issue 6, pp 1197–1238 | Cite as

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

  • Rafał Gruszczyński
  • Andrzej Pietruszczak
Open Access
Article

Abstract

This is the first, out of two papers, devoted to Andrzej Grzegorczyk’s point-free system of topology from Grzegorczyk (Synthese 12(2–3):228–235, 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.

Keywords

Grzegorczyk structures Point-free topology, Region-based topology Foundations of topology Mereology Mereological fields Mereological structures 

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© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of LogicNicolaus Copernicus University in ToruńToruńPoland

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