how do algebraic features of a set enrich its analytical and topological ones?

Question

I know this question is not very well-defined. My only excuse is that it is very general and observational.

If a set had only been studied for its analytical and topological properties, how might a binary operation defined on this set (which has not been studied before) enrich the previous knowledge? How does an algebraic structure of a set changes our view of it?

Answer 1

Also speaking very generally, an algebraic structure might induce a geometric one, hence also an analytical and topological one. A good example are algebraic structures on Lie algebras, which often correspond to geometric structures on Lie groups associated to these Lie algebras.

A good resource are the notes by Bill Goldman on Geometric structures on manifolds.