Let $\; \mathbf{G} = \langle G,\cdot,\cal{T}\hspace{.06 in}\rangle \;$ be a topological group.  $\;\;$  Let $\mathbf{e}$ be the identity element of $\langle G,\cdot \rangle$.
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Assume $\{\mathbf{e}\}$ is closed in $\langle G,\cal{T}\hspace{.06 in}\rangle$. $\;\;$ Then, I have managed to convince myself that:
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a.  $\;\;$ ZF proves that $\langle G,\cal{T}\hspace{.06 in}\rangle$ is regular Haudorff.
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b.  $\;\;$ ZF + (Dependent Choice) $\;$ proves that $\langle G,\cal{T}\hspace{.06 in}\rangle$ is completely regular.
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1.  $\;$ Are those right?
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2.  $\;$ Does ZF prove that $\langle G,\cal{T}\hspace{.06 in}\rangle$ is completely regular?
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3. $\;$ If no to 2,
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 does assuming one or more of following the suffice for ZF to conclude that $\langle G,\cal{T}\hspace{.06 in}\rangle$ is completely regular?
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(i) $\;$ $\mathbf{G}$ is [two-sided complete](http://en.wikipedia.org/wiki/Uniform_space#Examples)
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(ii) $\;$  $\langle G,\cdot \rangle$ is abelian
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(iii) $\;$ Countable Choice