We construct a completely regular ordered space $(X,{\Cal T},\leq)$ such that $X$ is an $I$-space, the topology $\Cal T$ of $X$ is metrizable and the bitopological space $(X,{\Cal T}^\sharp,{\Cal T}^{\flat})$ is pairwise regular, but not pairwise completely regular. (Here ${\Cal T}^\sharp$ denotes the upper topology and ${\Cal T}^\flat$ the lower topology of $X$.)
@article{118718,
author = {Hans-Peter A. K\"unzi and Stephen W. Watson},
title = {A metrizable completely regular ordered space},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {35},
year = {1994},
pages = {773-778},
zbl = {0812.54038},
mrnumber = {1321247},
language = {en},
url = {http://dml.mathdoc.fr/item/118718}
}
Künzi, Hans-Peter A.; Watson, Stephen W. A metrizable completely regular ordered space. Commentationes Mathematicae Universitatis Carolinae, Tome 35 (1994) pp. 773-778. http://gdmltest.u-ga.fr/item/118718/
Bitopological spaces, Proc. London Math. Soc. 13 (1963), 71-89. (1963) | MR 0143169 | Zbl 0107.16401
Completely regular ordered spaces, Order 7 (1990), 283-293. (1990) | MR 1113204
Quasi-uniform spaces - eleven years later, Top. Proc. 18 (1993), to appear. | MR 1305128
Bitopological spaces and quasi-uniform spaces, Proc. London Math. Soc. 17 (1967), 241-256. (1967) | MR 0205221 | Zbl 0152.21101
Order and strongly sober compactifications, in: Topology and Category Theory in Computer Science, ed. G.M. Reed, A.W. Roscoe and R.F. Wachter, Clarendon Press, Oxford, 1991, pp. 179-205. | MR 1145775 | Zbl 0745.54012
Topology and Order, D. van Nostrand, Princeton, 1965. | MR 0219042 | Zbl 0333.54002
Ordered topological spaces and the representation of distributive lattices, Proc. London Math. Soc. 24 (1972), 507-530. (1972) | MR 0300949 | Zbl 0323.06011
Is every partially ordered space with a completely regular topology already a completely regular partially ordered space?, Math. Nachr. 161 (1993), 199-201. (1993) | MR 1251017