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
I have found an important paper ``Each regular paratopological group is completely regular'' by Taras Banakh and Alex Ravsky. Theorem 1 of this paper is provable in ZF+DC; however, it can fail in a model for ZF. Therefore, it cannot be used in ZF to prove that topological groups are completely regular in ZF. Therefore, this nice paper by Taras Banakh and Alex Ravsky does not solve my problem.
I thank you for your answer. Yes, following Pontryagin's book, I assume that all topolocgical groups are T1-spaces, so they are Hausdorff in ZF. There is a wonderful paper ``Continuing horrors of topology without choice'' by C. Good and I. J. Tree (Topology Appl. 63 (1995)) in which a special model for ZF is applied to show in it even a Hausdorff compact, connected space X, consistiing of more than one point, such that every real-valued continuous function on X is constant. This is partly why I am not sure whether every topological group must be completely regular in every model for ZF in which it exists. Moreover, in all known proofs in ZFC of the complete regularity of topological groups, the axiom of dependent choices (DC) has been used (unfortunately, mostly probably unconsciously). It is known that DC is independent of ZF. So, my question is: is there a model for ZF+the negation of DC in which there is a Hausdorff topological group which is not completely regular?
Thank you for your kind answer. Unfortunately, I know too little about the group of homeomprphisms of X onto itself to be able to deduce whether this somehow strange X is embeddable in this group. Anyway, I shall try to think about this idea. I have also tried to work with free groups but without sufficient success.
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