I am looking for an example of a topological group with countable tightness with the property then it is not metrizable, but every countable subset is metrizable but I cannot construct an example.


This question is inspired by my earlier question
[when-is-the-topology-generated-by-countable-subsets](https://mathoverflow.net/questions/186169/when-is-the-topology-generated-by-countable-subsets)
where someone gave me an example of a topological **space** with all of the above properties, namely the 1-point-compactification of an uncountable discrete set and now I was wondering if there are examples of topological spaces with these properties that carry a group strucuture.


Thanks in advance, Tom



A countable tight topological group where every countable subset is metrizable
Actually, Santi Spadaro almost gave the answer in a comment to your previous question: Let $X$ be the $\Sigma$-product of $2^{\omega\_1}$ (where $2=\{0,1\}$), that is, the subset of $2^{\omega\_1}$ of all points with at most countably many coordinates different from $0$. Of course $X$ is a subgroup of $2^{\omega\_1}$.


Then, as explained in Henno Brandsma's answer [here](https://mathoverflow.net/questions/78975/ccc-collectionwise-normality-paracompact/79021#79021), $X$ is Frechet-Urysohn (and thus countably tight) and non-paracompact (and thus non-metrizable). But any countable subset is contained in some $2^{\alpha}$ for a countable $\alpha$, and the latter is metrizable since it is homeomorphic to the Cantor cube $2^\omega$.


