On M1- and M3-properties in the setting of ordered topological spaces

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 M1and M3properties in the setting of ordered topological spaces. Among other results, we show that if (X,T,≤) is an M1 ordered topological Cand I-space then the bitopological space (X,T,T) is pairwise M1. Here, T := {U ∈ T |U is an upper set} and T := {L ∈ T |L is a lower set}.