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 Mi-spaces (i = 1, 2, 3) and established that metrizable ⇒ M1 ⇒ M2 ⇒ M3. He then asked whether these implications are reversible. Gruenhage [5] and Junnila [8] independently showed that M3 ⇒ M2. In this paper, we investigate the M1- and M3- properties in the setting of ordered topological spaces. Among other results, we show that if (X,T,≤) is an M1 ordered topological C- and I-space then the bitopological space (X,T♮,T♭) is pairwise M1. Here, $\mathcal{T}^\sharp :=\{U\in \tau | U\, \mbox{is an upper bound set}\}$ and $\mathcal{T}^\flat := \{ L | \, \mbox{is a lower set} \}$.