The embedding structure for linearly ordered topological spaces

TL;DR

This paper investigates the embedding structure of linearly ordered topological spaces (LOTS), establishing results about maximal elements, bases, and the internal structure of subclasses, with implications for both countable and uncountable LOTS.

Contribution

It provides new results on the existence of maximal elements and bases in the embedding quasi-order of LOTS, including under additional set-theoretic assumptions.

Findings

Countable LOTS have a maximal (universal) element and a finite basis.

Uncountable LOTS of size at least continuum have no maximal element and maximal dominating number.

Under Proper Forcing Axiom, there are finite bases for uncountable LOTS and dense uncountable LOTS.

Abstract

In this paper, the class of all linearly ordered topological spaces (LOTS) quasi-ordered by the embeddability relation is investigated. In ZFC it is proved that for countable LOTS this quasi-order has both a maximal (universal) element and a finite basis. For the class of uncountable LOTS of cardinality $\kappa$ it is proved that this quasi-order has no maximal element for $\kappa$ at least the size of the continuum and that in fact the dominating number for such quasi-orders is maximal, i.e. $2^\kappa$. Certain subclasses of LOTS, such as the separable LOTS, are studied with respect to the top and internal structure of their respective embedding quasi-order. The basis problem for uncountable LOTS is also considered; assuming the Proper Forcing Axiom there is an eleven element basis for the class of uncountable LOTS and a six element basis for the class of dense uncountable LOTS in which all points have countable cofinality and coinitiality.