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

Анотація

We prove that any topological group G containing a subspace X of the Sorgenfrey line has spread s(G)≤s(X×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 Lindelöf countable power and contains an uncountable subspace of the Sorgenfrey line. This cometrizable topological group has countable spread but is not cosmic.