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.

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Additional Information

DOI:
https://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