  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.
