The *Sorgenfrey line* $\mathbb S$ is the real line endowed with the topology generated by the base consisting of all half-intervals $[a,b)$ for real numbers $a<b$.

The Sorgenfrey line is first-countable and non-metrizable and hence is not homeomorphic to a topological group.

On the other hand, the Sorgenfrey line $\mathbb S$ is homeomorphic to a subset of a topological group. For example, the free topological group $F(\mathbb S)$ over $\mathbb S$ contains a closed topological copy of $\mathbb S$. But $F(\mathbb S)$ also contains a topological copy of the square $\mathbb S\times\mathbb S$ and hence $F(\mathbb S)$ contains an uncountable discrete subspace. Is this situation typical?

Problem.Let $G$ be a topological group containing a topological copy of the Sorgenfrey line. Does $G$ necessarily contain a uncountable discrete subspace?

**Added in Edit.** The answer to this problem is affirmative under OCA (the Open Coloring Axiom), which follows from PFA (the Proper Forcing Axiom).

Theorem (OCA).Under OCA, a topological group $G$ has uncountable spread if $G$ contains a subset, homeomorphic to an uncountable subspace of the Sorgenfrey line.

Combined with the result of Gruenhage, this allows to prove the following characterization of cosmic groups:

Theorem (OCA).A cometrizable topological group is cosmic if and only if it has countable spread.

We recall that a topological space has a *countable spread* if it does not contain an uncountable discrete subspace.

A topological space $X$ is *cometrizable* if it admits a weaker metrizable topology such that each point $x\in X$ has a (not necessarily open) neighborhood base consisting of metrically closed sets.

The following theorem shows that the above OCA-theorem is not true in ZFC.

Theorem (CH).Under CH there exists a cometrizable topological group which contains an uncountable subset of the Sorgenfrey line but is hereditarily Lindelof and hence has countable spread.

*Proof.* In this paper Michael constructs a CH-example of an uncountable subspace $X$ of the Sorgenfrey line whose countable power $X^\omega$ is hereditarily Lindelof. The space $X$ can be embedded into a cometrizable Boolean topological group $G$ so that $X$ generates $G$. Then $G$ is hereditarily Lindelof, being the countable union of continuous images of finite powers of $X$.