Is there a group $G$ which is at the same time a (compact-Hausdorff)-ly generated weakly Hausdorff space (or short CGWH space) such that inverse and product are continuous maps and the space is not Hausdorff?
Note that the group multiplication is a map from $G \times G$ to $G$. In the sense of this question, $G \times G$ denotes the product in the category of CGWH spaces. It is not necessarily equipped with the usual product topology.
To give this question the correct color, I give this proper answer. The mathematical part of this answer is purely based on Tyrone's comments.
Yes, there is a CGWH group which is not Hausdorff. This is shown in Lamartin's article in Example 2.14.
The idea is the following: The forgetful functor from CGWH groups to CGWH spaces has a left adjoint $F$. Let $X$ be any non-Hausdorff CGWH space. By definition, $FX$ is a CGWH group. There is a natural continuous map $X\to FX$. This can be seen by explicitely constructing $FX$ or by taking the adjoint to the continuous group Homomorphism $\operatorname{id}: FX \to FX$. One can show that this map is a closed embedding. Therefore, $FX$ is not Hausdorff.
