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Let $\; \mathbf{G} = \langle G,\cdot,\cal{T}\hspace{.06 in}\rangle \;$ be a topological group. $\;\;$ Let $\mathbf{e}$ be the identity element of $\langle G,\cdot \rangle$.
  Assume $\{\mathbf{e}\}$ is closed in $\langle G,\cal{T}\hspace{.06 in}\rangle$. $\;\;$ Then, I have managed to convince myself that:

a. $\;\;$ ZF proves that $\langle G,\cal{T}\hspace{.06 in}\rangle$ is regular Haudorff.
b. $\;\;$ ZF + (Dependent Choice) $\;$ proves that $\langle G,\cal{T}\hspace{.06 in}\rangle$ is completely regular.

1. $\;$ Are those right?
   
   
    
2. $\;$ Does ZF prove that $\langle G,\cal{T}\hspace{.06 in}\rangle$ is completely regular?
   
   
    
3. $\;$ If no to 2,
   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?
   
   (i) $\;$ $\mathbf{G}$ is two-sided complete
   (ii) $\;$ $\langle G,\cdot \rangle$ is abelian
   (iii) $\;$ Countable Choice

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:

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

My questions are:

1. Are those right?
2. Does ZF prove that $\langle G,\cal{T}\;\rangle$ is completely regular?
3. 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? 
4. $\mathbf{G}$ is two-sided complete 
5. $\langle G,\cdot \rangle$ is abelian 
6. Countable Choice

Let $\; \mathbf{G} = \langle G,\cdot,\cal{T}\hspace{.06 in}\rangle \;$ be a topological group. $\;\;$ Let $\mathbf{e}$ be the identity element of $\langle G,\cdot \rangle$.
Assume $\{\mathbf{e}\}$ is closed in $\langle G,\cal{T}\hspace{.06 in}\rangle$. $\;\;$ Then, I have manged to convince myself that:

a. $\;\;$ ZF proves that $\langle G,\cal{T}\hspace{.06 in}\rangle$ is regular Haudorff.
b. $\;\;$ ZF + (Dependent Choice) $\;$ proves that $\langle G,\cal{T}\hspace{.06 in}\rangle$ is completely regular.

1. $\;$ Are those right?
   
   
   
2. $\;$ Does ZF prove that $\langle G,\cal{T}\hspace{.06 in}\rangle$ is completely regular?
   
   
   
3. $\;$ If no to 2,
   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?
   
   (i) $\;$ $\mathbf{G}$ is two-sided complete
   (ii) $\;$ $\langle G,\cdot \rangle$ is abelian
   (iii) $\;$ Countable Choice

Let $\; \mathbf{G} = \langle G,\cdot,\cal{T}\hspace{.06 in}\rangle \;$ be a topological group. $\;\;$ Let $\mathbf{e}$ be the identity element of $\langle G,\cdot \rangle$.
Assume $\{\mathbf{e}\}$ is closed in $\langle G,\cal{T}\hspace{.06 in}\rangle$. $\;\;$ Then, I have managed to convince myself that:

a. $\;\;$ ZF proves that $\langle G,\cal{T}\hspace{.06 in}\rangle$ is regular Haudorff.
b. $\;\;$ ZF + (Dependent Choice) $\;$ proves that $\langle G,\cal{T}\hspace{.06 in}\rangle$ is completely regular.

1. $\;$ Are those right?
   
   
   
2. $\;$ Does ZF prove that $\langle G,\cal{T}\hspace{.06 in}\rangle$ is completely regular?
   
   
   
3. $\;$ If no to 2,
   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?
   
   (i) $\;$ $\mathbf{G}$ is two-sided complete
   (ii) $\;$ $\langle G,\cdot \rangle$ is abelian
   (iii) $\;$ Countable Choice