\(D\)-completion, well-filterification and sobrification (Q6611781)

The objective of the present paper is to gain further insight into the relationships of the \(D\)-completion, well-filterification and sobrification of a \(T_0\) space. The standard constructions are presented and specific examples related to the basic notions worked out. Also, a characterization of the \(D\)-completion of a poset (endowed with the Scott topology) by so-called pre-\(C\)-compact elements is provided. However, the main results concern conditions which guarantee that the three mentioned completions coincide. \N\NIn particular, it is proved that the \(D\)-completion agrees with the sobrification for (1) any \(T_0\) space where \(\downarrow(A \cap W )\) is closed for any irreducible closed set \(A\) and upper set \(W\) (in the specialization order), (2) any locally compact \(T_0\) space where \(\downarrow(A \cap K)\) is closed for any closed \textit{KF}-set \(A\) and compact saturated set \(K\), and (3) any core-compact and join-continuous poset. Additionally, if a \(T_0\) space is first countable, then its well-filterification is shown to coincide with its sobrification.