Off - Diagonal

Experience shows that there is a strong parallel between metrization theory for compact spaces and for linearly ordered spaces in terms of diagonal conditions. Recent theorems of Gruenhage, Pelant, Kombarov, and Stepanova have described metrizability of compact (and related) spaces in terms of the offdiagonal behavior of those spaces, i.e., in terms of properties of X −∆. In this paper, we show that these off-diagonal results have no analogs for linearly ordered topological spaces by constructing a nonmetrizable, first countable LOTS X that is paracompact off of the diagonal, has a locally finite rectangular open cover of X − ∆, and admits a collection U of subsets of X − ∆ that is σ-locally finite in X −∆, covers X −∆, and consists of co-zero subsets of X. Provided b = ω1, our example contains a Lindelöf subspace Y that has a countable rectangular open cover of Y 2 −∆ and yet does not have a Gδ-diagonal, thereby answering a question of Kombarov. In addition, we consider the role of much stronger off-diagonal covering conditions such as the Lindelöf property and hereditary paracompactness. MR numbers (1991): 54F05, 54E35, 54G20