It seems to me that in a topological group $G$ if $H$ is a connected subset and $K$ is any subset then the subgroup generated by all commutators $hkh^{-1}k^{-1}$ is connected. $G$ does not to be Hausdorff, and $H$ and $K$ do not have to be closed or to be subgroups.

The ingredients in the proof are: (1) The image of a connected set under a continuous map is connected. (2) The union of a set of connected sets is connected if they all have some point in common. (3) The subgroup generated by a connected set is connected.

It seems to me that in a topological group $G$ if $H$ is a connected subset containing the identity and $K$ is any subset then the subgroup generated by all commutators $hkh^{-1}k^{-1}$ is connected. $G$ does not to be Hausdorff, and $H$ and $K$ do not have to be closed or to be subgroups.

The ingredients in the proof are: (1) The image of a connected set under a continuous map is connected. (2) The union of a set of connected sets is connected if they all have some point in common. (3) The subgroup generated by a connected set is connected.