# What is known about topological groups of countable spread in ZFC?

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", 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.

Problem. Is this corollary true in ZFC? In other words: has each topological group of countable spread countable pseudocharacter?

Remark. The Peng-Wu example of an L-group shows that topological groups with countable spread need not be separable.

## 1 Answer

The answer is no.

In the paper "A separable normal topological group need not be Lindelöf" (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 $$CH$$).

• Thank you very much for the answer. Maybe this group is a Borel image of a Polish space and hence it gives also a counterexample to this problem: mathoverflow.net/a/320637/61536 Or it is too optimistic? Commented Jan 18, 2019 at 20:32