On Meet-Continuous Dcpos

It is well-known that a complete lattice L is a meet-continuous lattice if and only if \(x \wedge \vee D = { \vee _{d \in D}}x \wedge d\) for all x ∈ P. This property in fact can be characterized by the Scott topology simply as clσ (↓x ∩ ↓D) = ↓x whenever x ≤ ∨ D. Since the meet operator is not involved, the topological property of meet-continuity can be naturally extended to general dcpos. Such dcpos are also called meet-continuous in this note. It turns out that there exist close relations among meet-continuity, Hausdorff separation, quasicontinuity, continuity and Scott-open filter bases. In particular, we prove that Hausdorff dcpos (via the Lawson topology) need not be quasicontinuous, the category CONT is not a reflective full subcategory of QCONT, the category of quasicontinuous domains, and a dcpo P is meet-continuous when \(\sigma (P) = \overline \sigma (P)\) or it is a semilattice with a \({T_0}\overline \sigma (D)\)-topology, where \(\overline \sigma (P)\) denote the topology generated by all the Scott-open filters of P. Moreover, under appropriate conditions, the category of dcpos with \({T_0}\overline \sigma (D)\)-topology form a cartesian closed category.