A metrizable semitopological semilattice with non-closed partial order - Banakh Taras; Bardyla Serhii; Ravsky Alex (Topological Algebra and its Applications, 2020)
We construct a metrizable semitopological semilattice X whose partial order P = {(x, y) ∈ X × X : xy = x} is a non-closed dense subset of X × X. As a by-product we find necessary and sufficient conditions for the existence of a (metrizable) Hausdorff topology on a set, act, semigroup or semilattice, having a prescribed countable family of convergent sequences.