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## Fundamenta Mathematicae

1997 | 153 | 2 | 99-123
Tytuł artykułu

### Diagonal conditions in ordered spaces

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Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
EN
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
99-123
Opis fizyczny
Daty
wydano
1997
otrzymano
1996-01-22
poprawiono
1996-10-13
Twórcy
autor
• Mathematics Department, Texas Tech University, Lubbock, Texas 79409, U.S.A.
autor
• Mathematics Department, College of William and Mary, Williamsburg, Virginia 23187, U.S.A.
Bibliografia
• [A] A. V. Arhangel'skii [A. V. Arkhangel'skiĭ], A survey of $C_p$-theory, Questions Answers Gen. Topology 5 (1987), 1-109.
• [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.
• [B1] H. Bennett, On quasi-developable spaces, Gen. Topology Appl. 1 (1971), 253-262.
• [B2] H. Bennett, Point-countability in linearly ordered spaces, Proc. Amer. Math. Soc. 28 (1971), 598-606.
• [Bo] C. Borges, On stratifiable spaces, Pacific J. Math. 17 (1966), 1-17.
• [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.
• [E] R. Engelking, General Topology, Heldermann, Berlin, 1989.
• [EL] R. Engelking and D. Lutzer, Paracompactness in ordered spaces, Fund. Math. 94 (1977), 49-58.
• [Fa] M. Faber, Metrizability in Generalized Ordered Spaces, Math. Centre Tracts 53, Mathematisch Centrum, Amsterdam, 1974.
• [Hc] S. Hechler, On the existence of certain cofinal subsets of $^ωω$, in: Proc. Sympos. Pure Math. 13, Amer. Math. Soc., Providence, R.I., 1974, 155-173.
• [H] H. Herrlich, Ordnungsfähigkeit total-diskontinuierlicher Räume, Math. Ann. 159 (1965), 77-80.
• [H1] M. Hušek, Continuous mappings on subspaces of products, in: Sympos. Math. 17, Academic Press, London, 1976, 25-41.
• [H2] M. Hušek, Topological spaces without κ-accessible diagonal, Comment. Math. Univ. Carolin. 18 (1977), 777-788.
• [JS] I. Juhász and Z. Szentmiklóssy, Convergent free sequences in compact spaces, Proc. Amer. Math. Soc. 116 (1992), 1153-1160.
• [L1] D. Lutzer, A metrization theorem for linearly orderable spaces, Proc. Amer. Math. Soc. 22 (1969), 557-558.
• [L2] D. Lutzer, On generalized ordered spaces, Dissertationes Math. 89 (1971).
• [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.
• [M2] E. Michael, Paracompactness and the Lindelöf property in finite and countable Cartesian products, Compositio Math. 23 (1971), 199-214.
• [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.
• [Ok] A. Okuyama, On metrizability of M-spaces, Proc. Japan Acad. 40 (1964), 176-179.
• [P] S. Purisch, Scattered compactifications and the orderability of scattered spaces, Proc. Amer. Math. Soc. 95 (1985), 636-640.
• [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).
• [So] R. Sorgenfrey, On the topological product of paracompact spaces, Bull. Amer. Math. Soc. 53 (1947), 631-632.
• [St] A. Stone, On σ-discreteness and Borel isomorphism, Amer. J. Math. 85 (1963), 655-666.
• [vW] J. van Wouwe, GO-Spaces and Generalizations of Metrizability, Math. Centre Tracts 104, Mathematisch Centrum, Amsterdam, 1979.
• [Zh] H. X. Zhou, On the small diagonals, Topology Appl. 13 (1982), 283-293.
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