Post History

Are we in a "history-valley" for Topology?

Here is my current rough timeline of Topology:

* Newton invents the calculus
* People like Riemann and Cauchy make it rigorous, and by this time, we have the $\varepsilon,\delta$ definition of continuity of a real valued function of a real variable.
* We realize that we want a definition of the continuity of vector-valued functions of vector spaces, possibly over complex numbers.  So we have to generalize the idea of a continuous function.  This makes it clear that we start from an open neighborhood of a point in the codomain and search for an appropriate open neighborhood in the domain.
* We start encountering other weird functions, for which we still want a notion of continuity.  From probability theory we want set functions.  From Fourier analysis we want maps from function spaces to function spaces.  So we need a broader notion of continuity.  We hone in on the idea of an open set not determined by any distance metric.  In order to do this we need the ability to declare the open sets in a way that replaces and generalizes our notion of an open ball.  
* Chaos reigns.  Different people start talking about a topology, but define it in different ways.  Some people include the Hausdorff axioms, some don't.  Some people define it this way, some people define it that way.  
* ???
* Now professors yell at you if you don't understand it fast enough because it's just a way of saying what your open sets are, what's so hard about that?!

So a few questions.  

1. Am I embarrassingly wrong about any of that?  

2. What happened in the "???" phase?  In particular, why did we eventually decide that we **wanted** the Sierpinski topology to be a topology, for instance?  Why did we ultimately not pack the Hausdorff axioms into the idea of a topology, and instead left it as a property that some topologies can have but others might not?  What is the value added by doing it this way?  Or to put it the other way around, would we be any worse off if we required topologies to satisfy properties that make it more intelligible from its growth out of the idea of open balls, and give a different name to structures that are merely "subsets of the powerset containing {}, closed under unions and countable intersections"?  

(This is the idea of a "history-valley" in my title: We got to a point where we do things this way, not because it's an optimum but because of a sequence of historically random events.  And now that we're here it's too hard to climb out.  Or at least the question is whether that's true, or is there a better motivation for why we do things the way we do them?)