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

Year 2015, Volume: 44 Issue: 6, 1391 - 1395, 01.12.2015

Hans-peter A. Künzi , Zechariah Mushaandja

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 $\mathcal{T}^\flat := \{ L | \, \mbox{is a lower set} \}$. 

Keywords

C-space, I-space, closure-preserving, (pairwise) M1, (pairwise) stratifiable

References

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