  We construct a metrizable semitopological semilattice $X$ whose partial order
$P=\{(x,y)\in X\times X:xy=x\}$ is a non-closed dense subset of $X\times X$. 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.
