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A note on paracompactness
in generalized ordered spaces


Authors: Eric K. van Douwen and David J. Lutzer
Journal: Proc. Amer. Math. Soc. 125 (1997), 1237-1245
MSC (1991): Primary 54F05, 54D20
DOI: https://doi.org/10.1090/S0002-9939-97-03902-6
MathSciNet review: 1396999
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we show that for generalized ordered spaces, paracompactness is equivalent to Property D, where a space $X$ is said to have Property D if, given any collection $\{G(x)\colon x\in X\}$ of open sets in $X$ satisfying $x\in G(x)$ for each $x$, there is a closed discrete subset $D$ of $X$ satisfying $X=\bigcup \{G(x)\colon x\in D\}$.


References [Enhancements On Off] (What's this?)

  • [BW] Carlos R. Borges and Albert C. Wehrly, A study of 𝐷-spaces, Topology Proc. 16 (1991), 7–15. MR 1206448
  • [vDP] Eric K. van Douwen and Washek F. Pfeffer, Some properties of the Sorgenfrey line and related spaces, Pacific J. Math. 81 (1979), no. 2, 371–377. MR 547605
  • [F] M. J. Faber, Metrizability in generalized ordered spaces, Mathematisch Centrum, Amsterdam, 1974. Mathematical Centre Tracts, No. 53. MR 0418053
  • [GH] Gillman, L., and Henricksen, M., Concerning rings of continuous functions, Trans. Amer. Math. Soc., 77 (1954), 340-362. MR 16:156g
  • [L1] D. J. Lutzer, On generalized ordered spaces, Dissertationes Math. Rozprawy Mat. 89 (1971), 32. MR 0324668
  • [L2] D. J. Lutzer, Ordinals and paracompactness in ordered spaces, TOPO 72—general topology and its applications (Proc. Second Pittsburgh Internat. Conf., Pittsburgh, Pa., 1972; dedicated to the memory of Johannes H. de Groot), Springer, Berlin, 1974, pp. 258–266, Lecture Notes in Math., Vol. 378. MR 0362250
  • [L3] David J. Lutzer, Ordered topological spaces, Surveys in general topology, Academic Press, New York-London-Toronto, Ont., 1980, pp. 247–295. MR 564104

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Additional Information

Eric K. van Douwen
Affiliation: Department of Mathematics, Ohio University, Athens, Ohio 45701

David J. Lutzer
Affiliation: Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187
Email: DJLUTZ@MAIL.WM.EDU

DOI: https://doi.org/10.1090/S0002-9939-97-03902-6
Keywords: Paracompactness, Property D, generalized ordered space, linearly ordered space
Received by editor(s): October 5, 1995
Communicated by: Franklin D. Tall
Article copyright: © Copyright 1997 American Mathematical Society