Summary: For a space \(X\) and a regular uncountable cardinal \(\kappa\leq | X|\) the authors say that \(\kappa\in D(X)\) if for each \(T\subset X^2-\Delta(X)\) with \(| T|=\kappa\), there is an open neighborhood \(W\) of \(\Delta(X)\) such that \(| T-W|=\kappa\). If \(\omega_1\in D(X)\) then they say that \(X\) has a small diagonal, and if every regular uncountable \(\kappa\leq | X|\) belongs to \(D(X)\) then they say that \(X\) has an H-diagonal.
In this paper the interplay between \(D(X)\) and topological properties of \(X\) in the category of generalized ordered spaces is investigated. The authors 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 . They give examples showing that their 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. Their example shows that a recent CH-result of I. Juhász and Z. Szentmiklóssy [Proc. Am. Math. Soc. 116, No. 4, 1153-1160 (1992; Zbl 0767.54002)] on metrizability of compact Hausdorff spaces with small diagonals cannot be generalized beyond the class of locally compact spaces. The authors present examples showing the interplay of the above diagonal conditions with set theory in a natural extension of the Michael line construction.