The structure of Valdivia compact lines

TL;DR

This paper characterizes Valdivia compact lines, disproves a previous conjecture about their structure, and refines conditions under which a compact line is Valdivia compact, including the importance of scattered closures.

Contribution

It provides an internal characterization of Valdivia compact lines and refines the conditions for their classification, correcting a prior conjecture with a counter-example.

Findings

Counter-example disproves the previous conjecture.

Valdivia compactness depends on the scatteredness of the closure of uncountable character points.

The conjecture holds if the closure of uncountable character points is scattered.

Abstract

We study linearly ordered spaces which are Valdivia compact in their order topology. We find an internal characterization of these spaces and we present a counter-example disproving a conjecture posed earlier by the first author. The conjecture asserted that a compact line is Valdivia compact if its weight does not exceed aleph one, every point of uncountable character is isolated from one side and every closed first countable subspace is metrizable. It turns out that the last condition is not sufficient. On the other hand, we show that the conjecture is valid if the closure of the set of points of uncountable character is scattered. This improves an earlier result of the first author.