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

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