A comparison of two systems of point-free topology
Keywords:point-free topology, region-based topology, foundations of topology, mereology, mereological structures, separation structures, connection structures, Grzegorczyk structures, Biacino-Gerla structures.
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 . In  Loredana Biacino and Giangiacomo Gerla presented an axiomatization which was inspired by the Grzegorczyk’s system, and which is its variation. Our aim is to compare the two approaches and show that they are slightly different. Except for pointing to dissimilarities, we also demonstrate that the theories coincide (in the sense that their axioms are satisfied in the same class of structures) in presence of axiom stipulating non-existence of atoms.
L. Biacino and G. Gerla, Connection structures: Grzegorczyk’s and Whitehead’s definitions of point, Notre Dame Journal of Formal Logic 37 (3) (1996), pp. 431–439.
R. Gruszczyński and A. Pietruszczak, Space, points and mereology. On foundations of point-free Euclidean geometry, Logic and Logical Philosophy 18 (2) (2009), pp. 145–188.
R. Gruszczyński and A. Pietruszczak, A study in Grzegorczyk point-free topology. Part I: Separation and Grzegorczyk structures, Studia Logica, Vol. 106 (2018), pp. 1197–1238.
R. Gruszczyński and A. Pietruszczak, A study in Grzegorczyk pointfree topology. Part II: Spaces of points, Studia Logica (2018).
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