Model-theoretic characterizations of arcs and simple closed curves

Author:
Paul Bankston

Journal:
Proc. Amer. Math. Soc. **104** (1988), 898-904

MSC:
Primary 03C20; Secondary 03C65, 54B25, 54D05, 54D35, 54F25, 54F65

MathSciNet review:
937843

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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.

**[1]**Paul Bankston,*Ultraproducts in topology*, General Topology and Appl.**7**(1977), no. 3, 283–308. MR**0458351****[2]**Paul Bankston,*Expressive power in first order topology*, J. Symbolic Logic**49**(1984), no. 2, 478–487. MR**745375**, 10.2307/2274179**[3]**Paul Bankston,*Reduced coproducts of compact Hausdorff spaces*, J. Symbolic Logic**52**(1987), no. 2, 404–424. MR**890449**, 10.2307/2274391**[4]**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**, 10.1016/0022-4049(90)90129-6**[5]**-,*Taxonomies of model-theoretically defined topological properties*(submitted).**[6]**C. C. Chang and H. J. Keisler,*Model theory*, North-Holland, Amsterdam, 1973.**[7]**N. Dunford and J. T. Schwartz,*Linear operators*, Part I, Interscience, New York, 1966.**[8]**R. Gurevič,*On ultracoproducts of compact Hausdorff spaces*, J. Symbolic Logic**53**(1988), no. 1, 294–300. MR**929393**, 10.2307/2274446**[9]**C. Ward Henson,*Nonstandard hulls of Banach spaces*, Israel J. Math.**25**(1976), no. 1-2, 108–144. MR**0461104****[10]**C. W. Henson, C. G. Jockusch Jr., L. A. Rubel, and G. Takeuti,*First order topology*, Dissertationes Math. (Rozprawy Mat.)**143**(1977), 40. MR**0432446****[11]**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****[12]**Stephen Willard,*General topology*, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1970. MR**0264581**

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1988-0937843-6

Keywords:
co-elementary equivalence,
compact Hausdorff spaces,
Peano continua,
ultracoproducts,
arcs,
simple closed curves

Article copyright:
© Copyright 1988
American Mathematical Society