# Diagonal conditions in ordered spaces

Fundamenta Mathematicae (1997)

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

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topBennett, 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 -

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