A topological space has *countable spread* if every discrete subspace is at most countable.


By Theorem 8.10 in Todorcevic's book "[Partition Problems in Topology](https://books.google.com.ua/books/about/Partition_Problems_in_Topology.html?id=k6UZnZ_qkgIC&redir_esc=y)", PFA implies that each regular space $X$ with countable spread is hereditarily Lindelof and hence has countable pseudocharacter (which means that each singleton $\{x\}$ in $X$ is a $G\_\delta$-set). This result entails the following 


**Corollary.** *Under PFA, each topological group of countable spread has countable pseudocharacter.*



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> **Problem.** Is this corollary true in ZFC? In other words: *has each topological group of countable spread countable pseudocharacter?*
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**Remark.** The [Peng-Wu example](https://settheory.mathtalks.org/wp-content/uploads/2016/12/Peng.pdf) of an L-group shows that topological groups with countable spread need not be separable.



What is known about topological groups of countable spread in ZFC?
The answer is no.


In the paper *"[A separable normal topological group need not be Lindelöf](https://www.sciencedirect.com/science/article/pii/0016660X76900337)"* (General topology and its applications, 1976), Hajnal and Juhász use the continuum hypothesis to construct a hereditarily separable topological group $G$ which is not Lindelöf. They construct it as a dense subgroup of $2^{\omega\_1}$ and it has many other properties (e.g. it is countably compact and hereditarily normal). It also has what they call in the paper property (P): for any $\alpha \in \omega\_1$ the restriction map $2^{\omega\_1} \to 2^\alpha$ maps $G$ onto the whole $2^\alpha$.


It is clear that hereditarily separable is stronger than countable spread and that property (P) implies uncountable pseudocharacter, so this $G$ is a counterexample for your corollary (under $CH$).


