A Study in Grzegorczyk Point-Free Topology Part II: Spaces of Points

Abstract

In the second installment to Gruszczyński and Pietruszczak (Stud Log, 2018. https://doi.org/10.1007/s11225-018-9786-8) we carry out an analysis of spaces of points of Grzegorczyk structures. At the outset we introduce notions of a concentric and $\omega$-concentric topological space and we recollect some facts proven in the first part which are important for the sequel. Theorem 2.9 is a strengthening of Theorem 5.13, as we obtain stronger conclusion weakening Tychonoff separation axiom to mere regularity. This leads to a stronger version of Theorem 6.10 (in form of Corollary 2.10). Further, we show that Grzegorczyk points are maximal contracting filters in the sense of De Vries (Compact spaces and compactifications, Van Gorcum and Comp. N.V., 1962), but the converse inclusion is not necessarily true. We also compare the notions of a Grzegorczyk point and an ultrafilter, and establish several properties of topological spaces based on Grzegorczyk structures. The main results of the paper are representation and completion theorems for G-structures. We prove both set-theoretical and topological representation theorems for various classes of G-structures. We also present topological object duality theorem for the class of complete G-structures and the class of concentric spaces, both restricted to structures which satisfy countable chain condition. We conclude the paper with proving equivalence of the original Grzegorczyk axiom with the one accepted by us as axiom (G).