D-completion, well-filterification and sobrification

TL;DR

This paper explores conditions under which D-completion, well-filterification, and sobrification coincide in T0 spaces, providing examples and new characterizations that advance understanding of these topological constructs.

Contribution

It establishes sufficient conditions for the equivalence of D-completion, well-filterification, and sobrification in T0 spaces and introduces a new characterization using pre-c-compact elements.

Findings

D-completion can coincide with well-filterification under certain conditions

An example shows tapered closed sets may not be closures of directed sets

A new characterization of D-completion via pre-c-compact elements

Abstract

In this paper, we obtain some sufficient conditions for the D-completion of a T0 space to be the well-filterification of this space, the well-filterification of a T0 space to be the sobrification of this space and the D-completion of a T0 space to be the sobrification, respectively. Moreover, we give an example to show that a tapered closed set may be neither the closure of a directed set nor the closed KF-set, respectively. Because the tapered closed set is a closed WD-set, the example also gives a negative answer to a problem proposed by Xu. Meantime, a new direct characterization of the D-completion of a T0 space is given by using the notion of pre-c-compact elements.