I know that It holds true in ZFC (even in ZF+DC) that every topological group is completely regular. It is also known to me that, in some models for ZF, a topology induced by a uniformity can be not completely regular. This is why, while working in a model for ZF in which AC fails, I cannot use the theorem of ZFC that every topology induced by a uniformity is completely regular to show that topological groups are completely regular in this model. I would be grateful if someone from the mathematical community could tell me whether a topological group can fail to be completely regular in a model for ZF or whether this is an open problem to the whole mathematical community. Of course, I assume that all topological groups are T1-spaces.
Regards, Eliza Wajch