# Is there a subject in mathematics like topological Algebra?

I would consider myself an algebraic topologist and there is a lot of influence from algebra into topology and without this input from the algebraic site I would say that a lot of topological theorems wouldn't been proved today.

On the other hand I only know of a few examples, where topology can prove an algebraic statement (for example that $\mathbb{Z}$ is the only discrete subgroup of $\mathbb{R}$ follows from the classification of $1$-dimensional manifolds; a lot of proofs of the fundamental theorem of algebra and so on) and I have never heard of an algebraic statement, where no purely algebraic proof exists, but a topological one.

So my question is are there algebraic statements that so far have only been proven in topological ways or is there even a whole field of mathematics called "topological algebra"?

• The first step in algebraic geometry is topologizing a ring('s set of prime ideals) and studying that space instead of the ring's elements themselves. Jun 26, 2016 at 13:09
• In Serre's Trees, he uses actions on simplicial trees to show that every subgroup of a free group is free, providing a topological proof of the Nielsen-Schreier theorem. More generally, this has developed into Bass-Serre theory. Jun 26, 2016 at 17:38
• You might be interested in the subject where "topological" is narrowed down to "geometric" and "algebra" is narrowed down to "group theory". "Geometric group theory" is a very big and exciting field of mathematics. It even has it's own tag here on math.stackexchange. Jun 26, 2016 at 19:02

A result I like, and the proof is accessible to an advnced undergraduate, is the classification of closed subgroups of the additive group $\mathbb R^n$. Any such is isomorphic to a product of copies of $\mathbb R$ and $\mathbb Z$. A good reference for this is Abelian Groups (London Mathematical Society Lecture Note Series) Paperback Aug 2008 by Sidney A. Morris.