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?

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**Added in Edit.** The answer to this problem is affirmative under OCA (the [Open Coloring Axiom][1]), 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][2], this allows to prove the following characterization of cosmic groups:

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

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

The proofs of these two results are not very short, so I will add a link to the proof as soon as the corresponding paper will be ready.

>So, now the question is about the answer to the above problem in ZFC. 
 


  [1]: https://en.wikipedia.org/wiki/Open_coloring_axiom
  [2]: http://www.ams.org/journals/tran/1989-313-01/S0002-9947-1989-0992600-5/S0002-9947-1989-0992600-5.pdf