How (or why) did Topology become so central to modern mathematics? It is frequently said that topology is nowadays one of the central pillars of modern mathematics (ex. "Because of its central place in a broad spectrum of mathematics")  The field has managed to blossom into a magnificently large field of mathematics, with tons of major applications into fields that even exist outside of math. How and why did this happen? Did topology originally come about as a way to solve unsolved problems at its time of origin, and later grew into what it is today? How has it become so central? (Side note: On a not too uncommon basis, topology seems to be one of the more 'quirky' fields of math.)
 A: One reason is that limits and/or convergence are very important, because they allow you to approximate complicated objects by simple objects. For example, maybe you want to approximate a function by polynomials (taylor series) or by sinusoids (fourier analysis). From this standpoint, the important observation is that every topology induces a convergence structure, and topologies were invented first. Keywords: functional analysis, topological vector space.
A second reason is that (classical) differential geometry is based on the concept of a manifold, which is usually formulated in a way that makes reference to topological spaces. Indeed, at the bottom rung of the manifold ladder are "topological manifolds", which can be defined as topological spaces that look locally like $\mathbb{R}^n$, subject to the some further constraints.
A third reason is that the basic definitions of homotopy theory make sense over general topological spaces; you don't need a manifold.
A fourth reason is the Zariski topology just so happens to be a topology.
A fifth reason is that often when you combine infinitely-many set-theoretic objects together in an interesting way, the resulting structure often ends up carrying a topology in a natural way. Look up "profinite completion of the integers." Its fundamental to modern number theory.
A sixth reason is that we can consider the sheaves on a topological space, and $\mathbf{Set}$-valued sheaves are about as fundamental of an object as there is, because they always form a topos, which is a place where higher-order intuitionistic logic can be interpreted.
A: First of all, even as a topologist, I wouldn't say that topology is any more central than a variety of other fields: analysis, algebra, and so on. Even then, there are areas of topology that really don't have any broad applications outside the subject itself: surgery theory, some of the very technical computations of the stable homotopy groups of spheres, etc. Topology at least appears in a variety of subjects, though, because it seems to be the suitable framework for a lot of math. In introductory real analysis, the intermediate value theorem is a statement about connectedness. The objects in differential geometry, analysis on manifolds, etc. are topological spaces (with a lot of extra structure). Even some constructions that aren't inherently topological, like (in)direct limits, naturality, often pop up first for students via algebraic topology classes. 
Here's a more specific example. One of the most familiar examples of a cohomology theory is de Rham cohomology on a smooth manifold. This is a very concrete object: It's the space of solutions to a certain differential equation, modulo a space of 'trivial' solutions. Those equations always have a local solution; the problem is that those local solutions don't necessarily patch together to give a global solution. Cohomology groups are the obstructions to this patching construction. The same idea of obstructions to solving a global problem locally pops up in several different areas of math. Sheaf cohomology on a scheme, for example, is quite concretely a similar sort of obstruction, even though the Zariski topology is very different from the topology of a smooth manifold (or even CW-complex). There's another kind of cohomology defined for groups acting on modules, the connection being that abstract nonsense gives a connection between the group cohomology of $\pi$ and the (say, singular) cohomology of $K(\pi, 1)$. (I'm skipping over some nontrivial details here.) The same idea pops up in complex analysis (the Cousin problems, for example), various parts of physics, etc.; furthermore, extensions of this idea pop up in algebraic geometry, $K$-theory, and so on.
A: In a very simplistic way we can think of it like this. Analysis means limit. Limit means continuity. Continuity means open set. And open sets mean topology. So in a way topology is the basis of analysis. Or at least that's how I see it.
