On $M_1$- and $M_3$-properties in the setting of ordered topological spaces

Abstract

In 1961, J. G. Ceder [3] introduced and studied classes of topological spaces called M i -spaces ( i = 1 , 2 , 3 ) and established that metrizable ⇒ M 1 ⇒ M 2 ⇒ M 3 . He then asked whether these implications are reversible. Gruenhage [5] and Junnila [8] independently showed that M 3 ⇒ M 2 . In this paper, we investigate the M 1 - and M 3 - properties in the setting of ordered topological spaces. Among other results, we show that if ( X, T , ≤ ) is an M 1 ordered topological C - and I -space then the bitopological space ( X, T ♮ , T ♭ ) is pairwise M 1 . Here, $\mathcal{T}^\sharp :=\{U\in \tau | U\, \mbox{is an upper bound set}\}$ and $\mathcal{T}^\flat := \{ L | \, \mbox{is a lower set} \}$.