A colleague and I have a result that shows that for Hausdorff countably compact W-spaces, being a topological group implies normality. But it occurred to us that (being not super experienced working with topological groups) we're unsure whether this is known (or disproven) without the W-space assumption. π-Base is also uncertain:
So, are Hausdorff countably compact topological groups always normal?
Best Answer
I haven't checked all details, but this should be a counter-example:
Let $\Sigma$ be the $\Sigma$-product of $ω_1$ copies of the circle group $\mathbb T$, considered as a subgroup of $\mathbb T^{\omega_1}$, $G = \Sigma \times \mathbb T^{\omega_1}$.
Of course, $G$ is a Hausdorff topological group. It is mentioned here, page 5.
According to this paper, $\Sigma$ is countably compact and $G$ is not normal. Since $\mathbb T^{\omega_1}$ is compact, $G$ is countably compact (Engelking, 3.10.14).
See here for the general definition of $\Sigma$-product and a proof that it is countably compact, if all factors are compact. Of course, in case of groups, the base point in each factor is the identity.