Constructing compacta from posets

• Autores: Adam Bartoš, Tristan Bice, Alessandro Vignati
• Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 69, Nº 1, 2025, págs. 217-265
• Idioma: inglés
• DOI: 10.5565/publmat6912510
• Enlaces
  • Texto completo
• Resumen

  • We develop a simple method of constructing topological spaces from countable posets with finite levels, one which applies to all second-countable T1 compacta. This results in a duality amenable to building such spaces from finite building blocks, essentially an abstract analogue of classical constructions defining compacta from progressively finer open covers.

• Referencias bibliográficas
  • J. W. Alexander, Ordered sets, complexes and the problem of compactification, Proc. Natl. Acad. Sci. USA 25(6) (1939), 296–298. DOI: 10.1073/pnas.25.6.296
  • A. Bartoˇs and W. Kubi´s, Hereditarily indecomposable continua as generic mathematical structures, Preprint (2022). arXiv:2208.06886.
  • M. Bell and J. Ginsburg, Compact spaces and spaces of maximal complete subgraphs, Trans.Amer. Math. Soc. 283(1) (1984), 329–338. DOI: 10.2307/2000007
  • T. Bice, Gr¨atzer–Hofmann–Lawson–Jung–S¨underhauf duality, Algebra Universalis 82(2) (2021), Paper no. 35, 13 pp. DOI: 10.1007/s00012-021-00729-2
  • T. Bice and W. Kubi´s, Wallman duality for semilattice subbases, Houston J. Math. 48(1) (2022), 1–31.
  • T. Bice and C. Starling, General non-commutative locally compact locally Hausdorff Stone duality, Adv. Math. 341 (2019), 40–91. DOI: 10.1016/j.aim.2018.10.031
  • R. H. Bing, A homogeneous indecomposable plane continuum, Duke Math. J. 15(3) (1948), 729–742. DOI: 10.1215/S0012-7094-48-01563-4
  • S. A. Celani and L. J. Gonzalez ´ , A categorical duality for semilattices and lattices, Appl. Categ. Structures 28(5) (2020), 853–875. DOI:...
  • W. Debski and E. D. Tymchatyn, Cell structures and topologically complete spaces, Topology Appl. 239 (2018), 293–307. DOI: 10.1016/j.topol.2018.02.020
  • L. B. Fuller, Trees and proto-metrizable spaces, Pacific J. Math. 104(1) (1983), 55–75. DOI: 10.2140/pjm.1983.104.55
  • S. J. van Gool, Duality and canonical extensions for stably compact spaces, Topology Appl. 159(1) (2012), 341–359. DOI: 10.1016/j.topol.2011.09.040
  • G. Grabner, Spaces having Noetherian bases, in: Proceedings of the 1983 Topology Conference (Houston, Tex., 1983), Topology Proc. 8(2) (1983),...
  • G. Gratzer ¨ , General Lattice Theory, Pure Appl. Math. 75, Academic Press, Inc. [HarcourtBrace Jovanovich, Publishers], New York-London,...
  • K. H. Hofmann and J. D. Lawson, The spectral theory of distributive continuous lattices, Trans. Amer. Math. Soc. 246 (1978), 285–310. DOI:...
  • T. Irwin and S. Solecki, Projective Fra¨ıss´e limits and the pseudo-arc, Trans. Amer. Math. Soc. 358(7) (2006), 3077–3096. DOI: 10.1090/S0002-9947-06-03928-6
  • A. Jung and P. Sunderhauf ¨ , On the duality of compact vs. open, in: Papers on General Topology and Applications (Gorham, ME, 1995), Ann....
  • T. Kawai, Predicative theories of continuous lattices, Log. Methods Comput. Sci. 17(2) (2021), Paper no. 22, 38 pp. DOI: 10.23638/LMCS-17(2:22)2021
  • J. Krasinkiewicz and P. Minc, Nonexistence of universal continua for certain classes of curves, Bull. Acad. Polon. Sci. S´er. Sci. Math. Astronom....
  • K. Kunen, Set Theory, Stud. Log. (Lond.) 34, College Publications, London, 2011.
  • E. E. Moise, An indecomposable plane continuum which is homeomorphic to each of its nondegenerate subcontinua, Trans. Amer. Math. Soc. 63...
  • C. Mummert and F. Stephan, Topological aspects of poset spaces, Michigan Math. J. 59(1) (2010), 3–24. DOI: 10.1307/mmj/1272376025
  • J. R. Munkres, Topology, Second edition, Prentice Hall, Inc., Upper Saddle River, NJ, 2000.
  • S. B. Nadler, Jr., Continuum Theory. An introduction, Monogr. Textbooks Pure Appl. Math. 158, Marcel Dekker, Inc., New York, 1992. DOI: 10.1201/9781315274089
  • L. G. Oversteegen and E. D. Tymchatyn, On hereditarily indecomposable compacta, in: Geometric and Algebraic Topology, Banach Center Publ....
  • J. Picado and A. Pultr, Frames and Locales. Topology without points, Front. Math., Birkh¨auser/Springer Basel AG, Basel, 2012. DOI: 10.1007/978-3-0348-0154-6
  • H. A. Priestley, Representation of distributive lattices by means of ordered stone spaces, Bull. London Math. Soc. 2(2) (1970), 186–190. DOI:...
  • I. F. Putnam, Cantor Minimal Systems, Univ. Lecture Ser. 70, American Mathematical Society, Providence, RI, 2018. DOI: 10.1090/ulect/070
  • N. Sauer and R. E. Woodrow, Finite cutsets and finite antichains, Order 1(1) (1984), 35–46. DOI: 10.1007/BF00396272
  • T. Shirota, A generalization of a theorem of I. Kaplansky, Osaka Math. J. 4(2) (1952), 121–132. DOI: 10.18910/10334
  • M. H. Stone, The theory of representations for Boolean algebras, Trans. Amer. Math. Soc. 40(1) (1936), 37–111. DOI: 10.2307/1989664
  • M. H. Stone, Topological representations of distributive lattices and Brouwerian logics, Casopis ˇPˇest. Mat. Fys. 67(1) (1938), 1–25.
  • H. de Vries, Compact spaces and compactifications. An algebraic approach, Thesis (Ph.D.)- Universiteit van Amsterdam (1962).
  • H. Wallman, Lattices and topological spaces, Ann. of Math. (2) 39(1) (1938), 112–126. DOI: 10.2307/1968717