I am studying topological groups, and I have been able to do quite a lot on my own by proving the propositions in this link on my own, but when I read up wikipedia that topological groups are all completely regular, I wasn't able to either find a proof or do it myself, which got me concerned.
The question is ; can anyone confirm me that topological groups are indeed completely regular, and if so, where could I find a proof, or do you know it?
One proof I would be interested in is that accordingly to this Wikipedia page concerning uniform spaces, a topological group can be equipped with the structure of a uniform space in a canonical way, and uniform spaces are completely regular. I don't know how to prove this one either.
I would also be interested in a more direct approach if possible for the purposes of giving a talk on topological spaces ; if a direct proof is shorter than the one going through uniform spaces, it would give me more time in my talk to mention other things.