Model-Theoretic Characterizations of Arcs and Simple Closed Curves

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. 0. Introduction. An arc is a topological space homeomorphic with the closed unit interval I on the real line; a simple closed curve (s.c.c.) is a space homeomorphic with the standard unit circle S in the plane. By a continuum we mean a connected compact Hausdorff space; a locally connected metrizable continuum is termed a Peano continuum. (Peano continua are precisely those Hausdorff spaces that are continuous images of I, by the Hahn-Mazurkiewicz theorem [12].) A simple triod is any space homeomorphic with the set { (x, y): 1 < x < 1 and y = 0, or x = 0 and 0 < y < 1} in the plane. A point of a connected space is a cut point if the complement of that point is disconnected. Our topological nomenclature is predominately from Willard [12]. Characterizations of I and S go back to the 1920's, principally to the work of R. L. Moore. 0.1 THEOREM (MOORE [12]). If X is a metrizable continuum with exactly two noncut points, then X is an arc. C1 0.2 THEOREM (MOORE [12]). If X is a nondegenerate metrizable continuum such that the complement of any two points is disconnected, then X is a s.c.c. C1 0.3 THEOREM (MOORE [11]). If X is a nondegenerate Peano continuum containing no simple triod, then X is either an arc or a s.c.c. 1I The first model-theoretic characterization of I and S of which we are aware appears in the paper [10] by C. W. Henson, C. J. Jockusch, L. A. Rubel, and G. Takeuti. For any space X, let F(X) be its bounded lattice of closed sets. We use standard model-theoretic terminology, as can be found in [6]. Received by the editors June 13, 1987 and, in revised form, April 6, 1988. Presented at the Annual Meeting of the Association for Symbolic Logic, New York City, December 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 03C20, 03C65, 54B25, 54D05, 54D35, 54F25, 54F65.