On the spread of topological groups containing subsets of the Sorgenfrey line

TL;DR

This paper investigates the spread of topological groups containing Sorgenfrey line subsets, establishing conditions under which such groups are cosmic or have uncountable spread, depending on set-theoretic assumptions.

Contribution

It proves a lower bound on the spread of topological groups with Sorgenfrey line subspaces and explores the impact of set-theoretic axioms on their properties.

Findings

Under OCA, groups with Sorgenfrey line subspaces have uncountable spread.

Under CH, there exists a cometrizable Abelian group with Sorgenfrey line subspace that has countable spread.

A cometrizable topological group can have countable spread but not be cosmic.

Abstract

We prove that any topological group $G$ containing a subspace $X$ of the Sorgenfrey line has spread $s(G)\ge s(X\times X)$. Under OCA, each topological group containing an uncountable subspace of the Sorgenfrey line has uncountable spread. This implies that under OCA a cometrizable topological group $G$ is cosmic if and only if it has countable spread. On the other hand, under CH there exists a cometrizable Abelian topological group that has hereditarily Lindel\"of countable power and contains an uncountable subspace of the Sorgenfrey line. This cometrizable topological group has countable spread but is not cosmic.