Diagonal conditions in ordered spaces

Harold Bennett; David Lutzer

Fundamenta Mathematicae (1997)

  • Volume: 153, Issue: 2, page 99-123
  • ISSN: 0016-2736

Abstract

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For a space X and a regular uncountable cardinal κ ≤ |X| we say that κ ∈ D(X) if for each T ⊂ X 2 - Δ ( X ) with |T| = κ, there is an open neighborhood W of Δ(X) such that |T - W| = κ. If ω 1 ∈ D ( X ) then we say that X has a small diagonal, and if every regular uncountable κ ≤ |X| belongs to D(X) then we say that X has an H-diagonal. In this paper we investigate the interplay between D(X) and topological properties of X in the category of generalized ordered spaces. We obtain cardinal invariant theorems and metrization theorems for such spaces, proving, for example, that a Lindelöf linearly ordered space with a small diagonal is metrizable. We give examples showing that our results are the sharpest possible, e.g., that there is a first countable, perfect, paracompact Čech-complete linearly ordered space with an H-diagonal that is not metrizable. Our example shows that a recent CH-result of Juhász and Szentmiklóssy on metrizability of compact Hausdorff spaces with small diagonals cannot be generalized beyond the class of locally compact spaces. We present examples showing the interplay of the above diagonal conditions with set theory in a natural extension of the Michael line construction.

How to cite

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Bennett, Harold, and Lutzer, David. "Diagonal conditions in ordered spaces." Fundamenta Mathematicae 153.2 (1997): 99-123. <http://eudml.org/doc/212222>.

@article{Bennett1997,
abstract = {For a space X and a regular uncountable cardinal κ ≤ |X| we say that κ ∈ D(X) if for each $T ⊂ \{X^2\} - Δ(X)$ with |T| = κ, there is an open neighborhood W of Δ(X) such that |T - W| = κ. If $ω_1 ∈ D(X)$ then we say that X has a small diagonal, and if every regular uncountable κ ≤ |X| belongs to D(X) then we say that X has an H-diagonal. In this paper we investigate the interplay between D(X) and topological properties of X in the category of generalized ordered spaces. We obtain cardinal invariant theorems and metrization theorems for such spaces, proving, for example, that a Lindelöf linearly ordered space with a small diagonal is metrizable. We give examples showing that our results are the sharpest possible, e.g., that there is a first countable, perfect, paracompact Čech-complete linearly ordered space with an H-diagonal that is not metrizable. Our example shows that a recent CH-result of Juhász and Szentmiklóssy on metrizability of compact Hausdorff spaces with small diagonals cannot be generalized beyond the class of locally compact spaces. We present examples showing the interplay of the above diagonal conditions with set theory in a natural extension of the Michael line construction.},
author = {Bennett, Harold, Lutzer, David},
journal = {Fundamenta Mathematicae},
keywords = {H-diagonal; small diagonal; linearly ordered topological space; generalized ordered space; cardinal invariant, metrizability; paracompact space; Čech-complete space; p-space, Michael line; Sorgenfrey line; σ-disjoint base; -disjoint base; -space; H-diagonal.; cardinal invariant; Čech-complete linearly ordered space; Michael line},
language = {eng},
number = {2},
pages = {99-123},
title = {Diagonal conditions in ordered spaces},
url = {http://eudml.org/doc/212222},
volume = {153},
year = {1997},
}

TY - JOUR
AU - Bennett, Harold
AU - Lutzer, David
TI - Diagonal conditions in ordered spaces
JO - Fundamenta Mathematicae
PY - 1997
VL - 153
IS - 2
SP - 99
EP - 123
AB - For a space X and a regular uncountable cardinal κ ≤ |X| we say that κ ∈ D(X) if for each $T ⊂ {X^2} - Δ(X)$ with |T| = κ, there is an open neighborhood W of Δ(X) such that |T - W| = κ. If $ω_1 ∈ D(X)$ then we say that X has a small diagonal, and if every regular uncountable κ ≤ |X| belongs to D(X) then we say that X has an H-diagonal. In this paper we investigate the interplay between D(X) and topological properties of X in the category of generalized ordered spaces. We obtain cardinal invariant theorems and metrization theorems for such spaces, proving, for example, that a Lindelöf linearly ordered space with a small diagonal is metrizable. We give examples showing that our results are the sharpest possible, e.g., that there is a first countable, perfect, paracompact Čech-complete linearly ordered space with an H-diagonal that is not metrizable. Our example shows that a recent CH-result of Juhász and Szentmiklóssy on metrizability of compact Hausdorff spaces with small diagonals cannot be generalized beyond the class of locally compact spaces. We present examples showing the interplay of the above diagonal conditions with set theory in a natural extension of the Michael line construction.
LA - eng
KW - H-diagonal; small diagonal; linearly ordered topological space; generalized ordered space; cardinal invariant, metrizability; paracompact space; Čech-complete space; p-space, Michael line; Sorgenfrey line; σ-disjoint base; -disjoint base; -space; H-diagonal.; cardinal invariant; Čech-complete linearly ordered space; Michael line
UR - http://eudml.org/doc/212222
ER -

References

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  1. [A] A. V. Arhangel’skii [A. V. Arkhangel’skiĭ], A survey of C p -theory, Questions Answers Gen. Topology 5 (1987), 1-109. 
  2. [AT] A. V. Arhangel’skii [A. V. Arkhangel’skiĭ] and V. V. Tkačuk [V. V. Tkachuk], Calibers and point-finite cellularity of the space C p ( X ) and some questions of S. Gul’ko and M. Hušek, Topology Appl. 23 (1986), 65-73. 
  3. [B1] H. Bennett, On quasi-developable spaces, Gen. Topology Appl. 1 (1971), 253-262. Zbl0222.54037
  4. [B2] H. Bennett, Point-countability in linearly ordered spaces, Proc. Amer. Math. Soc. 28 (1971), 598-606. Zbl0197.19101
  5. [Bo] C. Borges, On stratifiable spaces, Pacific J. Math. 17 (1966), 1-17. Zbl0175.19802
  6. [vD] E. K. van Douwen, The integers and topology, in: Handbook of Set-Theoretic Topology K. Kunen and J. Vaughan (eds.), North-Holland, Amsterdam, 1984, 111-168. 
  7. [E] R. Engelking, General Topology, Heldermann, Berlin, 1989. 
  8. [EL] R. Engelking and D. Lutzer, Paracompactness in ordered spaces, Fund. Math. 94 (1977), 49-58. Zbl0351.54014
  9. [Fa] M. Faber, Metrizability in Generalized Ordered Spaces, Math. Centre Tracts 53, Mathematisch Centrum, Amsterdam, 1974. Zbl0282.54017
  10. [Hc] S. Hechler, On the existence of certain cofinal subsets of ω ω , in: Proc. Sympos. Pure Math. 13, Amer. Math. Soc., Providence, R.I., 1974, 155-173. 
  11. [H] H. Herrlich, Ordnungsfähigkeit total-diskontinuierlicher Räume, Math. Ann. 159 (1965), 77-80. Zbl0136.19804
  12. [H1] M. Hušek, Continuous mappings on subspaces of products, in: Sympos. Math. 17, Academic Press, London, 1976, 25-41. 
  13. [H2] M. Hušek, Topological spaces without κ-accessible diagonal, Comment. Math. Univ. Carolin. 18 (1977), 777-788. Zbl0374.54035
  14. [JS] I. Juhász and Z. Szentmiklóssy, Convergent free sequences in compact spaces, Proc. Amer. Math. Soc. 116 (1992), 1153-1160. Zbl0767.54002
  15. [L1] D. Lutzer, A metrization theorem for linearly orderable spaces, Proc. Amer. Math. Soc. 22 (1969), 557-558. Zbl0177.50703
  16. [L2] D. Lutzer, On generalized ordered spaces, Dissertationes Math. 89 (1971). 
  17. [M1] E. Michael, The product of a normal space and a metric space need not be normal, Bull. Amer. Math. Soc. 69 (1963), 375-376. Zbl0114.38904
  18. [M2] E. Michael, Paracompactness and the Lindelöf property in finite and countable Cartesian products, Compositio Math. 23 (1971), 199-214. Zbl0216.44304
  19. [Ml] A. Miller, Special subsets of the real line, in: Handbook of Set-Theoretic Topology, K. Kunen and J. Vaughan (eds.), North-Holland, Amsterdam, 1984, 201-234. 
  20. [Ok] A. Okuyama, On metrizability of M-spaces, Proc. Japan Acad. 40 (1964), 176-179. Zbl0127.38702
  21. [P] S. Purisch, Scattered compactifications and the orderability of scattered spaces, Proc. Amer. Math. Soc. 95 (1985), 636-640. Zbl0597.54024
  22. [S] V. E. Šneider [V. E. Shneĭder], Continuous images of Suslin and Borel sets. Metrization theorems, Dokl. Akad. Nauk SSSR 50 (1945), 77-79 (in Russian). Zbl0061.39705
  23. [So] R. Sorgenfrey, On the topological product of paracompact spaces, Bull. Amer. Math. Soc. 53 (1947), 631-632. Zbl0031.28302
  24. [St] A. Stone, On σ-discreteness and Borel isomorphism, Amer. J. Math. 85 (1963), 655-666. Zbl0117.40103
  25. [vW] J. van Wouwe, GO-Spaces and Generalizations of Metrizability, Math. Centre Tracts 104, Mathematisch Centrum, Amsterdam, 1979. Zbl0438.54030
  26. [Zh] H. X. Zhou, On the small diagonals, Topology Appl. 13 (1982), 283-293. Zbl0495.54028

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