Diagonal conditions in ordered spaces

Bennett, Harold ; Lutzer, David

Fundamenta Mathematicae, Tome 154 (1997), p. 99-123 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 1997-01-01

Zbl 0894.54025

EUDML-ID : urn:eudml:doc:212222

Bibliographie

[00000] [A] A. V. Arhangel’skii [A. V. Arkhangel’skiĭ], A survey of $C_p$-theory, Questions Answers Gen. Topology 5 (1987), 1-109.

[00001] [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.

[00002] [B1] H. Bennett, On quasi-developable spaces, Gen. Topology Appl. 1 (1971), 253-262. | Zbl 0222.54037

[00003] [B2] H. Bennett, Point-countability in linearly ordered spaces, Proc. Amer. Math. Soc. 28 (1971), 598-606. | Zbl 0197.19101

[00004] [Bo] C. Borges, On stratifiable spaces, Pacific J. Math. 17 (1966), 1-17. | Zbl 0175.19802

[00005] [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.

[00006] [E] R. Engelking, General Topology, Heldermann, Berlin, 1989.

[00007] [EL] R. Engelking and D. Lutzer, Paracompactness in ordered spaces, Fund. Math. 94 (1977), 49-58. | Zbl 0351.54014

[00008] [Fa] M. Faber, Metrizability in Generalized Ordered Spaces, Math. Centre Tracts 53, Mathematisch Centrum, Amsterdam, 1974. | Zbl 0282.54017

[00009] [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.

[00010] [H] H. Herrlich, Ordnungsfähigkeit total-diskontinuierlicher Räume, Math. Ann. 159 (1965), 77-80. | Zbl 0136.19804

[00011] [H1] M. Hušek, Continuous mappings on subspaces of products, in: Sympos. Math. 17, Academic Press, London, 1976, 25-41.

[00012] [H2] M. Hušek, Topological spaces without κ-accessible diagonal, Comment. Math. Univ. Carolin. 18 (1977), 777-788. | Zbl 0374.54035

[00013] [JS] I. Juhász and Z. Szentmiklóssy, Convergent free sequences in compact spaces, Proc. Amer. Math. Soc. 116 (1992), 1153-1160. | Zbl 0767.54002

[00014] [L1] D. Lutzer, A metrization theorem for linearly orderable spaces, Proc. Amer. Math. Soc. 22 (1969), 557-558. | Zbl 0177.50703

[00015] [L2] D. Lutzer, On generalized ordered spaces, Dissertationes Math. 89 (1971).

[00016] [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. | Zbl 0114.38904

[00017] [M2] E. Michael, Paracompactness and the Lindelöf property in finite and countable Cartesian products, Compositio Math. 23 (1971), 199-214. | Zbl 0216.44304

[00018] [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.

[00019] [Ok] A. Okuyama, On metrizability of M-spaces, Proc. Japan Acad. 40 (1964), 176-179. | Zbl 0127.38702

[00020] [P] S. Purisch, Scattered compactifications and the orderability of scattered spaces, Proc. Amer. Math. Soc. 95 (1985), 636-640. | Zbl 0597.54024

[00021] [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). | Zbl 0061.39705

[00022] [So] R. Sorgenfrey, On the topological product of paracompact spaces, Bull. Amer. Math. Soc. 53 (1947), 631-632. | Zbl 0031.28302

[00023] [St] A. Stone, On σ-discreteness and Borel isomorphism, Amer. J. Math. 85 (1963), 655-666. | Zbl 0117.40103

[00024] [vW] J. van Wouwe, GO-Spaces and Generalizations of Metrizability, Math. Centre Tracts 104, Mathematisch Centrum, Amsterdam, 1979. | Zbl 0438.54030

[00025] [Zh] H. X. Zhou, On the small diagonals, Topology Appl. 13 (1982), 283-293. | Zbl 0495.54028