A metrizable semitopological semilattice with non-closed partial order (Q2182485)

A semilattice is a poset \((X,\leq)\) such that \(x\wedge y:=\inf\{x,y\}\) exists for any \(x,y\in X\). A (semi)topological semilattice is a semilattice \(X\) endowed with a topology such that \(X\times X\ni (x,y)\mapsto x\wedge y\) is (separately) continuous. Obviously for any Hausdorff topological semilattice the partial order \(\leq_X:=\{(x,y)\in X\times X: x\leq y\}\) is closed in \(X\times X\). The authors show that this is not any more true for semitopological semilattices. They construct a metrizable semitopological semilattice \(X\) such that the partial order \(\leq_X\) is a non-closed dense subset of \(X\times X\). For that the authors first study for a topological space \((X,\tau)\) conditions under which for a given map \(\ell:\text{dom}(\ell)\rightarrow X\) defined on \(\text{dom}(\ell)\subseteq X^\omega\) any sequence \(s\in\text{dom}(\ell)\) converges to \(\ell(s)\) in \((X,\tau)\).