153 (1997)
Diagonal conditions in ordered spaces
by
Harold R. B e n n e t t (Lubbock, Tex.) and David J. L u t z e r (Williamsburg, Va.)
Dedicated to the memory of Maarten Maurice
Abstract. For a space X and a regular uncountable cardinal κ ≤ |X| we say that κ ∈ D(X) if for each T ⊂ X
2− ∆(X) with |T | = κ, there is an open neighborhood W of
∆(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 Lindel¨of 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 Cech-complete linearly ordered space with an H-diagonal that is not metrizable. Our ˇ example shows that a recent CH-result of Juh´asz and Szentmikl´ossy 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.
1. Introduction. As part of his study of functions defined on product spaces, M. Huˇsek introduced a family of diagonal conditions in a topological space X ([H1, H2]). For a cardinal number κ, he defined that the diagonal
∆(X) of a space X is κ-inaccessible if for any T ⊂ X
2− ∆(X), |T | = κ implies that |T − W | = κ for some open neighborhood W of ∆(X). If the diagonal of X is ω
1-inaccessible, he said that X has a small diagonal.
Subsequently, these notions have played a role in metrization theory ([Zh, JS]) and in the study of C
p(X) ([A, AT]), often for compact spaces, and
1991 Mathematics Subject Classification: Primary 54F05; 54D15; 54D30; Secondary 54E18; 54E35.
Key words and phrases: H-diagonal, small diagonal, linearly ordered topological space, generalized ordered space, cardinal invariant, metrizability, paracompact space, ˇ Cech- complete space, p-space, Michael line, Sorgenfrey line, σ-disjoint base.
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