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.
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Bennett, B., and I. Düntsch, Axioms, algebras and topology, in M. Aiello, I. Pratt-Hartmann, and J. van Benthem, (eds.), Handbook of Spatial Logics, Springer, 2007, pp. 99–159. https://doi.org/10.1007/978-1-4020-5587-4_3.
Biacino, L., and G. Gerla, Connection structures: Grzegorczyk’s and Whitehead’s definitions of point, Notre Dame J. of Formal Logic 37(3): 431–439, 1996. https://doi.org/10.1305/ndjfl/1039886519.
Dorais, F. G., Intersections of families of open sets ordered by well-inside relation in Euclidean space, MathOverflow (26.03.2016). http://mathoverflow.net/q/234572.
Düntsch, I., and M. Winter, A representation theorem for Boolean contact algebras, Theoretical Computer Science (B) 347(3): 498–512, 2005. https://doi.org/10.1016/j.tcs.2005.06.030.
Engelking, R., General Topology, revised and completed edition, Heldermann Verlag, Berlin, 1989. The first English edition: R. Engelking, General Topology, PWN, Warszawa, 1977. The Polish edition: R. Engelking, Topologia ogólna, PWN, Warszawa, 1976.
Gruszczyński, R., Niestandardowe teorie przestrzeni (Non-standard theories of space; in Polish), Nicolaus Copernicus University Scientific Publishing House, Toruń, 2016.
Gruszczyński, R., and A. Pietruszczak, Space, points and mereology. On foundations of point-free Euclidean geometry, Logic and Logical Philosophy 18(2): 145–188, 2009. https://doi.org/10.12775/LLP.2009.009.
Gruszczyński, R., and A. Pietruszczak, The relations of supremum and mereological sum in partially ordered sets, in C. Calosi and P. Graziani, (eds.), Mereology and the Sciences. Parts and Wholes in the Contemporary Scientific Context, Springer, 2014, pp. 123–140. https://doi.org/10.1007/978-3-319-05356-1_6.
Grzegorczyk, A., The systems of Leśniewski in relation to contemporary logical research, Studia Logica 3: 77–97, 1955. https://doi.org/10.1007/BF02067248.
Grzegorczyk, A., Axiomatizability of geometry without points, Synthese 12(2–3): 228–235, 1960. https://doi.org/10.1007/BF00485101.
Halmos, P., Naive Set Theory, Springer, New York, 1974.
Hamkins, J. D., and D. Seabold, Well-founded Boolean ultrapowers as large cardinal embeddings, 2012. http://arxiv.org/abs/1206.6075.
Koppelberg, S., Elementary arithmetic, in J. D. Monk and R. Bonnet, (eds.), Handbook of Boolean Algebras, volume 1, Elsevier, 1989, chapter 1, pp. 5–46.
Leonard, H. S., and N. Goodman, The calculus of individuals and its uses, Journal of Symbolic Logic 5(2): 45–55, 1940. https://doi.org/10.2307/2266169.
Pietruszczak, A., Metamereologia (Metamereology; in Polish), Nicolaus Copernicus University Press, Toruń, 2000.
Pietruszczak, A., Pieces of mereology, Logic and Logical Philosophy 14: 345–450, 2005. https://doi.org/10.12775/LLP.2005.014.
Pietruszczak, A., Podstawy teorii części (Foundations of the theory of parthood; in Polish), Nicolaus Copernicus University Scientific Publishing House, Toruń, 2013.
Pietruszczak, A., Classical mereology is not elementarily axiomatizable, Logic and Logical Philosophy 24(4): 485–498, 2015. https://doi.org/10.12775/LLP.2015.017.
Popvassilev, S. G., A question concerning Tychonoff spaces, Mathematics Stack Exchange (15.02.2017). http://math.stackexchange.com/q/2144869.
Roeper, P., Region-based topology, Journal of Philosophical Logic 26(3): 251–309, 1997. https://doi.org/10.1023/A:1017904631349.
Steen, L. A., and J. A. Seebach, Jr., Counterexamples in Topology, Springer-Verlag New York Inc, 1978.
Tarski, A., Foundations of the geometry of solids, in Logic, Semantics, Metamathematics. Papers from 1923 to 1938, Oxford University Press, 1956a, pp. 24–29.
Tarski, A., On the foundations of Boolean algebra, In Logic, Semantics, Metamathematics. Papers from 1923 to 1938, Oxford University Press, 1956b. pp. 320–341.
Vakarelov, D., Region-based theory of space: Algebras of regions, representation theory, and logics, in D. M. Gabbay, M. Zakharyaschev, S. S. Goncharov, (eds.), Mathematical Problems from Applied Logic II: Logics for the XXIst Century, Springer, New York, NY, 2007, pp. 267–348.
Whitehead, A. N., Process and Reality. An Essay in Cosmology, 1st edition 1929, with corrections, D. R. Griffin and D. W. Sherburne (eds.), The Free Press, A Division of Macmillan Publishing CO., INC., Nowy York, 1978.
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Presented by Jacek Malinowski; Received July 27, 2017.
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Gruszczyński, R., Pietruszczak, A. A Study in Grzegorczyk Point-Free Topology Part I: Separation and Grzegorczyk Structures. Stud Logica 106, 1197–1238 (2018). https://doi.org/10.1007/s11225-018-9786-8
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DOI: https://doi.org/10.1007/s11225-018-9786-8