Let $\mathbf{G} = \langle G,\cdot,\mathcal{T}\;\rangle$ be a topological group. Let $\mathbf{e}$ be the identity element of $\langle G,\cdot \rangle$. Assume $\{\mathbf{e}\}$ is closed in $\langle G,\cal{T}\;\rangle$. Then, I have managed to convince myself that:

- ZF proves that $\langle G,\cal{T}\;\rangle$ is regular Haudorff.
- ZF + (Dependent Choice) proves that $\langle G,\cal{T}\;\rangle$ is completely regular.

My questions are:

- Are those right?
- Does ZF prove that $\langle G,\cal{T}\;\rangle$ is completely regular?
- If no to question 2, does assuming one or more of following suffice for ZF to conclude that $\langle G,\cal{T}\,\rangle$ is completely regular?
- $\mathbf{G}$ is two-sided complete
- $\langle G,\cdot \rangle$ is abelian
- Countable Choice