Model-theoretic characterizations of arcs and simple closed curves
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- by Paul Bankston PDF
- Proc. Amer. Math. Soc. 104 (1988), 898-904 Request permission
Abstract:
Two compact Hausdorff spaces are co-elementarily equivalent if they have homeomorphic ultracopowers; equivalently if their Banach spaces of continuous real-valued functions have isometrically isomorphic Banach ultrapowers (or, approximately satisfy the same positive-bounded sentences). We prove here that any locally connected compact metrizable space co-elementarily equivalent with an arc (resp. a simple closed curve) is itself an arc (resp. a simple closed curve). The hypotheses of metrizability and local connectedness cannot be dropped.References
- Paul Bankston, Ultraproducts in topology, General Topology and Appl. 7 (1977), no. 3, 283–308. MR 458351
- Paul Bankston, Expressive power in first order topology, J. Symbolic Logic 49 (1984), no. 2, 478–487. MR 745375, DOI 10.2307/2274179
- Paul Bankston, Reduced coproducts of compact Hausdorff spaces, J. Symbolic Logic 52 (1987), no. 2, 404–424. MR 890449, DOI 10.2307/2274391
- Paul Bankston, Co-elementary equivalence for compact Hausdorff spaces and compact abelian groups, J. Pure Appl. Algebra 68 (1990), no. 1-2, 11–26. Special issue in honor of B. Banaschewski. MR 1082776, DOI 10.1016/0022-4049(90)90129-6 —, Taxonomies of model-theoretically defined topological properties (submitted). C. C. Chang and H. J. Keisler, Model theory, North-Holland, Amsterdam, 1973. N. Dunford and J. T. Schwartz, Linear operators, Part I, Interscience, New York, 1966.
- R. Gurevič, On ultracoproducts of compact Hausdorff spaces, J. Symbolic Logic 53 (1988), no. 1, 294–300. MR 929393, DOI 10.2307/2274446
- C. Ward Henson, Nonstandard hulls of Banach spaces, Israel J. Math. 25 (1976), no. 1-2, 108–144. MR 461104, DOI 10.1007/BF02756565
- C. W. Henson, C. G. Jockusch Jr., L. A. Rubel, and G. Takeuti, First order topology, Dissertationes Math. (Rozprawy Mat.) 143 (1977), 40. MR 432446
- R. L. Moore, Foundations of point set theory, Revised edition, American Mathematical Society Colloquium Publications, Vol. XIII, American Mathematical Society, Providence, R.I., 1962. MR 0150722
- Stephen Willard, General topology, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1970. MR 0264581
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 104 (1988), 898-904
- MSC: Primary 03C20; Secondary 03C65, 54B25, 54D05, 54D35, 54F25, 54F65
- DOI: https://doi.org/10.1090/S0002-9939-1988-0937843-6
- MathSciNet review: 937843