Today the language of topological spaces via open sets is fundamental in many different areas of mathematics, and it is a bit mysterious that the same formalism successfully captures such a wide variety of behavior. I can think of several independent reasons to invent the definition of a topology, all of which would have been on mathematicians' radar screens around the time the definition was first being mulled over in the early 20th century:
- To provide a foundation for Klein's Erlangen program and Poincare's work on Betti numbers and the fundamental group
- To clarify the foundations of calculus, e.g. the role of compactness in the extreme value theorem
- To distinguish among various notions of convergence of functions (leading to functional analysis)
- To give meaning to arguments involving "generic" configurations in algebraic geometry
My understanding is that it took quite some time for the modern formalism of topological spaces to emerge, so I'm wondering what specific results or examples were most influential in its development? And which modern applications of the theory were only realized after it matured?