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sniffnoy(And this is totally not what I'm supposed to be thinking about right now.)
Edit Oct 22: Solved problem of uniform continuity for Roelcke uniformity (see bottom).
OK. Left uniformity, right uniformity. Left uniformity makes left-multiplication uniformly continuous, right uniformity makes right-multiplication uniformly continuous.
I had convinced myself earlier that each only made the other uniformly continuous if the two were the same, but now it seems to me that in fact the left uniformity *always* makes right-multiplication uniformly continuous (and vice versa)? This seems fishy.
It is of course true that if either makes multiplication as a whole uniformly continuous, then the group is balanced. That's not in question.
And I had thought that if the 2-sided uniformity made either left-multiplication or right-multiplication uniformly continuous, then the group was balanced, but now it looks to me like this too is always uniformly continuous? (And if multiplication as a whole is uniformly continuous, then, uh...).
I mean, the math seems pretty clear, but I can't shake the feeling I must be doing something wrong...
Well, I guess I just screwed up before -- and so did that guy asking that question on Math.SE? That's a little less expected.
Going back and manually rechecking things, here is how I currently understand things:
Left uniformity: Makes left-translation and right-translation uniformly continuous. Makes inversion uniformly continuous iff group is balanced; same with multiplication.
Right uniformity: Same.
Two-sided uniformity: Makes left-translation, right-translation, and inversion uniformly continuous. Makes multiplication uniformly continuous iff group is balanced.
Roelcke uniformity: Makes left-translation, right-translation, and inversion uniformly continuous. Makes multiplication uniformly continuous iff group is balanced.
-Harry
Edit Oct 22: Solved problem of uniform continuity for Roelcke uniformity (see bottom).
OK. Left uniformity, right uniformity. Left uniformity makes left-multiplication uniformly continuous, right uniformity makes right-multiplication uniformly continuous.
I had convinced myself earlier that each only made the other uniformly continuous if the two were the same, but now it seems to me that in fact the left uniformity *always* makes right-multiplication uniformly continuous (and vice versa)? This seems fishy.
It is of course true that if either makes multiplication as a whole uniformly continuous, then the group is balanced. That's not in question.
And I had thought that if the 2-sided uniformity made either left-multiplication or right-multiplication uniformly continuous, then the group was balanced, but now it looks to me like this too is always uniformly continuous? (And if multiplication as a whole is uniformly continuous, then, uh...).
I mean, the math seems pretty clear, but I can't shake the feeling I must be doing something wrong...
Well, I guess I just screwed up before -- and so did that guy asking that question on Math.SE? That's a little less expected.
Going back and manually rechecking things, here is how I currently understand things:
Left uniformity: Makes left-translation and right-translation uniformly continuous. Makes inversion uniformly continuous iff group is balanced; same with multiplication.
Right uniformity: Same.
Two-sided uniformity: Makes left-translation, right-translation, and inversion uniformly continuous. Makes multiplication uniformly continuous iff group is balanced.
Roelcke uniformity: Makes left-translation, right-translation, and inversion uniformly continuous. Makes multiplication uniformly continuous iff group is balanced.
-Harry
