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

  • Rafał Gruszczyński
  • Andrzej Pietruszczak
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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).

Keywords

Grzegorczyk structures Point-free topology Region-based topology Mereology Mereological fields Mereological structures Representation theorems Regular spaces Concentric spaces 

Notes

Acknowledgements

Rafał Gruszczyński’s work was supported by National Science Center, Poland, grant Applications of mereology in systems of pointfree geometry, No. 2014/13/B/HS1/00766.

References

  1. 1.
    Biacino, L., and G. Gerla, Connection structures: Grzegorczyk’s and Whitehead’s definitions of point, Notre Dame Journal of Formal Logic 37(3): 431–439, 1996.  https://doi.org/10.1305/ndjfl/1039886519.CrossRefGoogle Scholar
  2. 2.
    De Vries, H., Compact spaces and compactifications, Van Gorcum and Comp. N.V., 1962.Google Scholar
  3. 3.
    Dimov, G., and D. Vakarelov, Contact algebras and region-based theory of space: A proximity approach-I, Fundamenta Informaticae 74(2–3): 209–249, 2006.Google Scholar
  4. 4.
    Gruszczyński, R., Niestandardowe teorie przestrzeni (Non-standard theories of space; in Polish), Nicolaus Copernicus University Scientific Publishing House, Toruń, 2016.Google Scholar
  5. 5.
    Gruszczyński, R., and A. Pietruszczak, A study in Grzegorczyk point-free topology Part I: Separation and Grzegorczyk structures, Studia Logica, 2018.  https://doi.org/10.1007/s11225-018-9786-8.Google Scholar
  6. 6.
    Grzegorczyk, A., Axiomatizability of geometry without points, Synthese 12(2–3): 228–235, 1960.  https://doi.org/10.1007/BF00485101.CrossRefGoogle Scholar
  7. 7.
    Hamkins, J. D., and D. Seabold, Well-founded Boolean ultrapowers as large cardinal embeddings, 2012. http://arxiv.org/abs/1206.6075.
  8. 8.
    Helmos, P. R., Lectures on Boolean Agebras, D. Van Nostrad Company, Inc., Princeton, New Jersey, Toronto, New York, London, 1963.Google Scholar
  9. 9.
    Koppelberg, S., Handbook of Boolean Algebras, vol. 1, J. D. Monk and R. Bonnet (eds.), Elsevier, 1989.Google Scholar
  10. 10.
    Pietruszczak, A., Podstawy teorii czȩści (Foundations of the theory of parthood; in Polish), Nicolaus Copernicus University Scientific Publishing House, Toruń, 2013.Google Scholar
  11. 11.
    Pietruszczak, A., Metamereology, Nicolaus Copernicus University Scientific Publishing House, Toruń, 2018.  https://doi.org/10.12775/9751.
  12. 12.
    Roeper, P., Region-based topology, Journal of Philosophical Logic 26(3): 251–309, 1997.  https://doi.org/10.1023/A:1017904631349.CrossRefGoogle Scholar

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© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of LogicNicolaus Copernicus University in ToruńToruńPoland

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