Topological ordered \(C\)- (resp. \(I\)-)spaces and generalized metric spaces (Q1032932)

Let \((X,\tau,\leq)\) be a topological ordered space. A subset \(A\) of \(X\) is an upper set (a lower set) if \(x\leq y\) and \(x\in A\) (\(x \geq y\) and \(x\in A\)) imply \(y\in A\). For \(E\subset X\), \(i(E)\) (\(d(E)\)) denotes the intersection of all upper (lower) sets of \(X\) containing \(E\). \textit{H. A. Priestley} [Proc. Lond. Math. Soc. 24, 507--530 (1972; Zbl 0323.06011)] defined \(X\) to be a \(C\)-space (\(I\)-space) if \(d(A)\) and \(i(A)\) are closed (open) sets whenever \(A\) is closed (open) in \(X\). The authors study properties of the bitopological space \((X,\tau^{\ell},\tau^u)\) associated with \((X,\tau,\leq)\), when \(X\) is either a \(C\)-space or an \(I\)-space (or both). Here \(\tau^{\ell}\) (\(\tau^u\)) is the collection of all \(\tau\)-open lower (upper) sets in \(X\). In particular, quasi-pseudometrizability of \((X,\tau^{\ell},\tau^u)\) is investigated.