A non-homogeneous zero-dimensional $X$ such that $X \times X$ is a group
HTML articles powered by AMS MathViewer
- by Fons van Engelen
- Proc. Amer. Math. Soc. 124 (1996), 2589-2598
- DOI: https://doi.org/10.1090/S0002-9939-96-03561-7
- PDF | Request permission
Abstract:
We provide an example of a zero-dimensional (separable metric) absolute Borel set $X$ which is not homogeneous, but whose square $X \times X$ admits the structure of a topological group. We also construct a zero-dimensional absolute Borel set $Y$ such that $Y$ is a homogeneous non-group but $Y \times Y$ is a group. This answers questions of Arhangel’skiĭ and Zhou.References
- Fredric D. Ancel and S. Singh, Rigid finite-dimensional compacta whose squares are manifolds, Proc. Amer. Math. Soc. 87 (1983), no. 2, 342–346. MR 681845, DOI 10.1090/S0002-9939-1983-0681845-X
- Fredric D. Ancel, Paul F. Duvall, and S. Singh, Rigid $3$-dimensional compacta whose squares are manifolds, Proc. Amer. Math. Soc. 88 (1983), no. 2, 330–332. MR 695269, DOI 10.1090/S0002-9939-1983-0695269-2
- A. V. Arhangel′skiĭ, The structure and classification of topological spaces and cardinal invariants, Uspekhi Mat. Nauk 33 (1978), no. 6(204), 29–84, 272 (Russian). MR 526012
- Fons van Engelen, Homogeneous Borel sets of ambiguous class two, Trans. Amer. Math. Soc. 290 (1985), no. 1, 1–39. MR 787953, DOI 10.1090/S0002-9947-1985-0787953-3
- F. van Engelen, Homogeneous zero-dimensional absolute Borel sets, CWI Tract 27, 1986.
- Fons van Engelen, On Borel groups, Topology Appl. 35 (1990), no. 2-3, 197–107. MR 1058800, DOI 10.1016/0166-8641(90)90105-B
- F. van Engelen, Zero-dimensional Borel groups of ambiguous class two, Erasmus University Report 9017/B (1990).
- Fons van Engelen and Jan van Mill, Borel sets in compact spaces: some Hurewicz type theorems, Fund. Math. 124 (1984), no. 3, 271–286. MR 774518, DOI 10.4064/fm-124-3-270-286
- Fons van Engelen, Arnold W. Miller, and John Steel, Rigid Borel sets and better quasi-order theory, Logic and combinatorics (Arcata, Calif., 1985) Contemp. Math., vol. 65, Amer. Math. Soc., Providence, RI, 1987, pp. 199–222. MR 891249, DOI 10.1090/conm/065/891249
- F. van Engelen, On Borel ideals, Ann. Pure Appl. Logic 70 (1994), 177-203.
- F. van Engelen, Boolean operations on Borel Wadge classes, in preparation.
- R. Engelking, W. Holsztyński and R. Sikorski, Some examples of Borel sets, Coll. Math. 15 (1966), 271–274.
- M. Lavrentieff, Sur les sous-classes de la classification de M. Baire, C. R. Acad. Sc. Paris 180 (1925), 111–114.
- A. Louveau, Some results in the Wadge-hierarchy of Borel sets, Cabal Seminar 79–81, Lect. Notes in Math. 1019 (1983), 28–55.
- A. Louveau and J. Saint-Raymond, Borel classes and closed games: Wadge-type and Hurewicz-type results, Trans. Amer. Math. Soc. 304 (1987), no. 2, 431–467. MR 911079, DOI 10.1090/S0002-9947-1987-0911079-0
- Jan van Mill, A rigid space $X$ for which $X\times X$ is homogeneous; an application of infinite-dimensional topology, Proc. Amer. Math. Soc. 83 (1981), no. 3, 597–600. MR 627701, DOI 10.1090/S0002-9939-1981-0627701-2
- John R. Steel, Analytic sets and Borel isomorphisms, Fund. Math. 108 (1980), no. 2, 83–88. MR 594307, DOI 10.4064/fm-108-2-83-88
- H. X. Zhou, Homogeneity properties and power spaces,, Ph. D. thesis, Wesleyan University, Middletown 1993.
Bibliographic Information
- Fons van Engelen
- Affiliation: Erasmus Universiteit, Econometrisch Instituut, Postbus 1738, 3000 DR Rotterdam, The Netherlands
- Email: engelen@wis.few.eur.nl
- Received by editor(s): February 20, 1995
- Communicated by: Franklin D. Tall
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 124 (1996), 2589-2598
- MSC (1991): Primary 54H05, 54E35, 54F65; Secondary 03E15
- DOI: https://doi.org/10.1090/S0002-9939-96-03561-7
- MathSciNet review: 1346990