Suddenly, I am confused about topological groups

(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

Quick note on equivalence of games (in Conway's sense)

So, Conway-style games. They have this weird equivalence relation on them which we always mod out by (ignoring set-theoretic issues :P ).

I don't know -- I feel like every time I've seen this stuff presented, the equivalence relation was just defined and presented, and it was just sort of taken for granted that yup, this is a meaningful equivalence relation. But that's never been clear to me. I mean obviously if you just want to do surreal numbers or nimbers or something you need to mod out by it, no question, there I have no trouble accepting it (although for those cases you could just consider nimbers as ordinals, or use an alternate construction of the real numbers). But for actual games?

I mean, consider the following: Games fall into four "win types" -- positive (left player wins), negative (right player wins), zero (second player wins), and fuzzy (first player wins). There are, under this equivalence relation, lots of positive, negative, and fuzzy games... but only one zero game. Doesn't that seem weird? I mean, like, what is this equivalence relation actually recording? Apparently all second-player-wins-games are *so* similar that no matter how different their internal structure, they all count as the same, but this doesn't hold true for games of other win types.

Anyway, the other day I finally came up with a sensible interpretation for this equivalence relation. This may be entirely standard, but nobody had ever bothered to mention it to me before, so, yeah.

The thing is that we have to also consider addition of games -- when we're considering games, their addition is a fundamental (and meaningful!) thing we want to consider. So what "A is equivalent to B" means is, for all games C, A+C and B+C have the same win type. And now it makes sense. (And then yay, the partial ordering and the multiplication and everything all respect this equivalence relation.)

Though it still seems entirely plausible to me that there might be stuff in combinatorial game theory you might want to study that doesn't respect this equivalence relation.

-Harry