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"?
 A: 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. 
A: 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.
