On Meet-Continuous Dcpos

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