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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 $X\times X$. Asa by-product we find necessary and sufficient conditions for the existence of a(metrizable) Hausdorff topology on a set, act, semigroup or semilattice, havinga prescribed countable family of convergent sequences.

A metrizable semitopological semilattice with non-closed partial order