Guessing clubs for aD, non D-spaces

TL;DR

This paper constructs a specific topological space demonstrating that being aD does not imply being linearly D, using Shelah's club guessing principles to answer a question posed by Arhangel'skii.

Contribution

It provides the first example of a 0-dimensional, scattered T2 space that is aD but not linearly D, advancing understanding of D-space properties.

Findings

Existence of a 0-dimensional, scattered T2 space that is aD but not linearly D

Use of Shelah's club guessing principles in topological space construction

Answers a longstanding question of Arhangel'skii

Abstract

We prove that there exists a 0-dimensional, scattered $T_2$ space $X$ such that $X$ is aD but not linearly D, answering a question of Arhangel'skii. The constructions are based on Shelah's club guessing principles.