Some further results on order-convergence in posets (Q1928414)

This paper deals with the question of when order-convergence in a poset is topological. The authors show that if order-convergence is topological in a poset then the poset is \(\mathcal{O}\)-doubly continuous. A poset is \(\mathcal{O}\)-doubly continuous if for each \(y\in P\), \(\{x\in P\mid x\ll_{\mathcal{O}}y\}\) is directed, \(\{x\in P\mid x\vartriangleright_{\mathcal{O}}y\}\) is filtered and \(\sup\{x\in P\mid x\ll_{\mathcal{O}}y\}=y=\inf\{x\in P\mid x\vartriangleright_{\mathcal{O}}y\}\). The relations \(x\ll_{\mathcal{O}}y\) and \(x\vartriangleright_{\mathcal{O}}y\) are defined as follows: Let \((y_{i})_{i\in I}\) be a net in \(P\) which order-converges to \(y,\) then \(x\ll_{\mathcal{O}}y\) (\(x\vartriangleright_{\mathcal{O}}y\)) if eventually \(y_{i}\geq x\) (\(y_{i}\leq x).\) The paper concludes with a necessary and sufficient condition (Condition \(\Delta\): In \(P,\;x\ll_{\mathcal{O}}y\leq z\) implies \(x\ll_{\mathcal{O}}z\) and \(s\vartriangleright_{\mathcal{O}}t\geq u\) implies \(s\vartriangleright_{\mathcal{O}}u\)), for order-convergence to be topological in the case of \(\mathcal{O}\)-doubly continuous posets.