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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"?

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    $\begingroup$ 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. $\endgroup$ Jun 26, 2016 at 13:09
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    $\begingroup$ 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. $\endgroup$ Jun 26, 2016 at 17:38
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    $\begingroup$ 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. $\endgroup$
    – Lee Mosher
    Jun 26, 2016 at 19:02

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There are indeed books on topological groups, Lie groups, topological rings, topological groupoids, Lie groupoids, .... , Lie double groupoids, ...

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.

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  • $\begingroup$ But every fact I know about Lie groups, topological algebraic objects and so on is a result about their topological structure our at least it is intertwined with the topological question. My question is if there is a purely algebraic fact, that has been proven by topological means. $\endgroup$
    – ThorbenK
    Jun 27, 2016 at 4:20
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There is an extensive literature which studies topological algebras from the perspective of universal algebra. Among other things, it focuses on the question of how algebraic structure restricts the nature of compatible topologies on the underlying space (e.g. -- a T_0 topological group must be completely regular). Walter Taylor has studied this extensively. Here is a nice example of what is known in the area.

In a slightly different direction, there have been a lot of generalizations of Stone Duality which shows that, in certain contexts, the distinction between algebra and topology is only one of perspective. Brian Davey is probably the foremost authority on generalizations of Stone Duality to other algebraic amd/or ordered structures, with his book (coauthored with David Clark) "Natural Dualities for the Working Algebraist" a nice introduction to the field.

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