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

A comparison of two systems of point-free topology

“…Strictly speaking, the axiom introduced by us is different from that presented in[5]. However, in[4] we demonstrated that both axiomatizations are equivalent to each other.…”

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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.

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“…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.…”

Mathematical methods in region-based theories of space: the case of Whitehead points

“…Secondly, it is evident that if a topological space satisfies the countable chain condition, then its algebra of regular open sets must also satisfy it. 15 The proof of this fact can be found in (Gruszczyński, 2016) and (Gruszczyński and Pietruszczak, 2019).…”

“…As we already wrote above, in RO(R n ) the set of Grzegorczyk points coincide with the set of Whitehead points, and therefore there are concrete contact algebras with Whitehead points. Moreover, thanks to (Gruszczyński, 2016) and (Gruszczyński and Pietruszczak, 2019) we know that every Grzegorczyk algebra gives rise to a concetric topological space. 20 Since Grzegorczyk points are points of the concentric space for RO(R n ), we know that there are indeed spaces whose points are Whitehead's constructs from Process and reality.…”