Electronic Version 22 (1997), 37-58

Harold Bennett and David Lutzer

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 off-diagonal behavior of those spaces, i.e., in terms of properties of X² - \Delta. In this paper, we show that these off-diagonal results have no analogs for linearly ordered topological spaces by constructing a non-metrizable, first countable LOTS X that is paracompact off of the diagonal, has a locally finite rectangular open cover of X² - \Delta, and admits a collection U of subsets of X² - \Delta that is \sigma-locally finite in X² - \Delta, covers X² - \Delta, and consists of co-zero subsets of X². Provided b = \omega₁, our example contains a Lindelöf subspace Y that has a countable rectangular open cover of Y² -\Delta and yet does not have a G_\delta-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.

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