Embedding partially ordered spaces in topological semilattices
Author: Lloyd D. Tucker
Journal: Proc. Amer. Math. Soc. 33 (1972), 203-206
MSC: Primary 06A10; Secondary 06A20
MathSciNet review: 0292724
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Abstract: A partial order on a compact space S is called continuous if is a closed subset of . In this paper, we define and study an embedding of the arbitrary compact continuously partially ordered space into a corresponding compact topological semilattice . We show that the structure of entirely determines the structure of . We prove that the inverse images under of components in are the order components of , where elements a and b of S are defined to be in the same order component of if there exists no continuous monotonic map which separates a and b. Finally, we show that is connected if and only if has only one order component.
-  Ernest Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152–182. MR 0042109, https://doi.org/10.1090/S0002-9947-1951-0042109-4
-  Leopoldo Nachbin, Sur les espaces topologiques ordonnés, C. R. Acad. Sci. Paris 226 (1948), 381–382 (French). MR 0023516
-  Lloyd D. Tucker, Generalized components and continuous orders (to appear).
-  L. E. Ward Jr., Concerning Koch’s theorem on the existence of arcs, Pacific J. Math. 15 (1965), 347–355. MR 0181981
- E. A. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152-182. MR 13, 54. MR 0042109 (13:54f)
- L. Nachbin, Sur les espaces topologiques ordonnés, C. R. Acad. Sci. Paris 226 (1948), 381-382; English transl. in Topology and order, Van Nostrand Math. Studies, Van Nostrand, Princeton, N.J., 1965. MR 9, 367; MR 36 #2125. MR 0023516 (9:367b)
- Lloyd D. Tucker, Generalized components and continuous orders (to appear).
- L. E. Ward, Jr., Concerning Koch's theorem on the existence of arcs, Pacific J. Math. 15 (1965), 347-355. MR 31 #6206. MR 0181981 (31:6206)
Keywords: Continuous orders, embedding, topological semilattice, order component, order connected
Article copyright: © Copyright 1972 American Mathematical Society