Abstract

AbstractThe purpose of this paper is to compare the notion of a Grzegorczyk point introduced in [19] (and thoroughly investigated in [3, 14, 16, 18]) to the standard notions of a filter in Boolean algebras and round filter in Boolean contact algebras. In particular, we compare Grzegorczyk points to filters and ultrafilters of atomic and atomless algebras. We also prove how a certain extra axiom influences topological spaces for Grzegorczyk contact algebras. Last but not least, we do not refrain from a philosophical interpretation of the results from the paper.

Highlights

  • In [16, 18] we carried out an extensive analysis of one of the first systems of point-free topology by Grzegorczyk [19], based on the notion of separation

  • A particular idiosyncrasy of Grzegorczyk’s approach to point-free topology is his definition of a point, which is a formal reflection of the geometrical intuition of a point as a system of “shrinking” regions of space

  • One of the problems that occupied us in the aforementioned works was the relation of Grzegorczyk points to other classical point-like constructions, such as ultrafilters and maximal round filters

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Summary

Introduction

In [16, 18] we carried out an extensive analysis of one of the first systems of point-free topology by Grzegorczyk [19], based on the notion of separation (the dual notion of contact). 1. In any complete BQCA, every G-point being an ultrafilter is principal and generated by an atom, i.e., Gpt ∩ Ult ⊆ PFAt. 2. In any BCA, every principal filter generated by an atom is a G-point being an ultrafilter and belongs to M.Rnd, i.e., PFAt ⊆ Gpt ∩ Ult ∩ M.Rnd = Gpt ∩ Ult. we have: Corollary 6.4.

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