Co-elementary equivalence, co-elementary maps, and generalized arcs

Bankston, Paul

doi:10.1090/S0002-9939-97-04088-4

Co-elementary equivalence, co-elementary maps, and generalized arcs
HTML articles powered by AMS MathViewer

by Paul Bankston

Proc. Amer. Math. Soc. 125 (1997), 3715-3720

DOI: https://doi.org/10.1090/S0002-9939-97-04088-4

PDF | Request permission

Abstract:

By a generalized arc we mean a continuum with exactly two non-separating points; an arc is a metrizable generalized arc. It is well known that any two arcs are homeomorphic (to the real closed unit interval); we show that any two generalized arcs are co-elementarily equivalent, and that co-elementary images of generalized arcs are generalized arcs. We also show that if $f:X \to Y$ is a function between compacta and if $X$ is an arc, then $f$ is a co-elementary map if and only if $Y$ is an arc and $f$ is a monotone continuous surjection.

References

B. Banaschewski, More on compact Hausdorff spaces and finitary duality, Canad. J. Math. 36 (1984), no. 6, 1113–1118. MR 771930, DOI 10.4153/CJM-1984-063-6
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, Model-theoretic characterizations of arcs and simple closed curves, Proc. Amer. Math. Soc. 104 (1988), no. 3, 898–904. MR 937843, DOI 10.1090/S0002-9939-1988-0937843-6
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
Paul Bankston, Taxonomies of model-theoretically defined topological properties, J. Symbolic Logic 55 (1990), no. 2, 589–603. MR 1056373, DOI 10.2307/2274649
———–, “On the topological model theory of normal disjunctive lattices,” (submitted)
C. C. Chang and H. J. Keisler, Model theory, 3rd ed., Studies in Logic and the Foundations of Mathematics, vol. 73, North-Holland Publishing Co., Amsterdam, 1990. MR 1059055
P. Eklof, “Ultraproducts for algebraists,” in “Handbook of Mathematical Logic,” North Holland, Amsterdam, 1977, pp. 105–137.
R. Gurevič, On ultracoproducts of compact Hausdorff spaces, J. Symbolic Logic 53 (1988), no. 1, 294–300. MR 929393, DOI 10.2307/2274446
John G. Hocking and Gail S. Young, Topology, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1961. MR 0125557
J. Nikiel, H. M. Tuncali, and E. D. Tymchatyn, Continuous images of arcs and inverse limit methods, Mem. Amer. Math. Soc. 104 (1993), no. 498, viii+80. MR 1148017, DOI 10.1090/memo/0498
J. Rosický, “Categories of models,” Seminarberichte Mathematik Informatik Fernuniversität 19(1984), 337–413.
H. Wallman, “Lattices and topological spaces,” Ann. Math.(2) 39(1938), 112–126.
Stephen Willard, General topology, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1970. MR 0264581