A metrizable semitopological semilattice with non-closed partial order

Abstract

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.

Keywords

• semitopological semilattice
• partial order
• convergent sequence
• act
• semigroup
• 54a20
• 06a12
• 22a26
• 37b05