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just for completeness, I have created an entry almost connected topological group.
It says the “quotient topological space $G/G_0$” and it means the quotient topological group. I know, you wanted to say the underlying space of the quotient topological group, but if one talks about quotient topological space then this means that only $G_0$ is contracted to a point, while the rest stays the same. In other words, the notation $G/G_0$ in topological category is much bigger than the underlying space of $G/G_0$ in the category of topological groups. The quotient in the category of topological spaces is different than the underlying topological space of a quotient in the category of topological groups. I corrected the statement.
But a real question is if the entry assumes Hausdorfness. I wonder how one can have compact quotient of topological groups by a connected component if not finite, as if something is an accumulation point, by the homogeneity, everything would be an accumulation point. Example ?
Zoran, as an example, consider the $p$-adic integers under addition. This is totally disconnected, so the connected component $G_0$ is just the identity element. Therefore $G/G_0 \cong G$, and $G$ is compact Hausdorff, so this is an example.
Thanks.
This example is mentioned already at maximal compact subgroup. But it would be nice if somebody found the time to add it also to almost connected topological group, highlighting the point just discussed.
(I guess I could do it later. But I won’t protest if somebody does it before me ;-)
The example if $p$-adic integers isn’t in either article. And I can’t decide where to put it in either article.
And I can’t decide where to put it in either article.
Maybe we should splitt off an entry p-adic integer from p-adic number anyway. The non almost-connectedness would naturally be discussed there and then we could point to that from the other two entries.
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