Direct sum of topological groups

In mathematics, a topological group G is called the topological direct sum[1] of two subgroups H₁ and H₂ if the map

{\begin{aligned}H_{1}\times H_{2}&\longrightarrow G\\(h_{1},h_{2})&\longmapsto h_{1}h_{2}\end{aligned}}

is a topological isomorphism.

More generally, G is called the direct sum of a finite set of subgroups $H_{i},i=1,\ldots ,n$ of the map

{\begin{aligned}\prod _{i=1}^{n}H_{i}&\longrightarrow G\\(h_{i})_{i\in I}&\longmapsto h_{1}h_{2}\cdots h_{n}\end{aligned}}

Note that if a topological group G is the topological direct sum of the family of subgroups $H_{i}$ then in particular, as an abstract group (without topology) it is also the direct sum (in the usual way) of the family $H_{i}$ .

Topological direct summands

Given a topological group G, we say that a subgroup H is a topological direct summand of G (or that splits topologically from G) if and only if there exist another subgroup K ≤ G such that G is the direct sum of the subgroups H and K.

A the subgroup H is a topological direct summand if and only if the extension of topological groups

0\to H{\stackrel {i}{{}\to {}}}G{\stackrel {\pi }{{}\to {}}}G/H\to 0

splits, where $i$ is the natural inclusion and $\pi$ is the natural projection.

Examples

Suppose that $G$ is a locally compact abelian group that contains the unit circle $\mathbb {T}$ as a subgroup. Then $\mathbb {T}$ is a topological direct summand of G. The same assertion is true for the real numbers $\mathbb {R}$ [2]

gollark: The problem with spaces is that you can’t actually see them. So you can’t be sure they’re correct. Also they aren’t actually there anyway - they are the absence of code. “Anti-code” if you will. Too many developers format their code “to make it more maintainable” (like that’s actually a thing), but they’re really just filling the document with spaces. And it’s impossible to know how spaces will effect your code, because if you can’t see them, then you can’t read them. Real code wizards know to just write one long line and pack it in tight. What’s that you say? You wrote 600 lines of code today? Well I wrote one, and it took all week, but it’s the best. And when I hand this project over to you next month I’ll have solved world peace in just 14 lines and you will be so lucky to have my code on your screen <ninja chop>.

gollark: Remove the call stack and do trampolining or something?

gollark: Yes, I think this is possible.

gollark: (ethically)

gollark: I might convert you into muons.

References

E. Hewitt and K. A. Ross, Abstract harmonic analysis. Vol. I, second edition, Grundlehren der Mathematischen Wissenschaften, 115, Springer, Berlin, 1979. MR0551496 (81k:43001)
Armacost, David L. The structure of locally compact abelian groups. Monographs and Textbooks in Pure and Applied Mathematics, 68. Marcel Dekker, Inc., New York, 1981. vii+154 pp. ISBN 0-8247-1507-1 MR0637201 (83h:22010)

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[1] E. Hewitt and K. A. Ross, Abstract harmonic analysis. Vol. I, second edition, Grundlehren der Mathematischen Wissenschaften, 115, Springer, Berlin, 1979. MR0551496 (81k:43001)

[2] Armacost, David L. The structure of locally compact abelian groups. Monographs and Textbooks in Pure and Applied Mathematics, 68. Marcel Dekker, Inc., New York, 1981. vii+154 pp. ISBN 0-8247-1507-1 MR0637201 (83h:22010)