TITLE: Why should algebraic objects have naturally associated topological spaces? (Formerly: What is a topological space?)
QUESTION [38 upvotes]: In this question, Harry Gindi states:

The fact that a commutative ring has a natural topological space associated with it is a really interesting coincidence.

Moreover, in the answers, Pete L. Clark gives a list of other "really interesting coincidences" of algebraic objects having naturally associated topological spaces.
Is there a deeper explanation of the occurrence of these "really interesting coincidences"? It seems to suggest that the standard definition of "topological space" (collection of subsets, unions, intersections, blah blah), which somehow always seemed kind of a weird and artificial definition to me, has some kind of deeper significance or explanation, since it pops up everywhere...
The (former) title of this question is meant to be provocative ;-)

See also:
What are interesting families of subsets of a given set? 
How can I really motivate the Zariski topology on a scheme? --- particularly Allen Knutson's answer

Edit 1: I should clarify a bit. Let me be more explicit: Is there a unified explanation (mathematical ... or perhaps not) for why various algebraic (where "algebraic" is loosely defined) objects should have naturally associated topological spaces? Pete in the comments notes that he does not like the use of the word "coincidence" here --- but if these things are not coincidences, then what's the explanation?
Of course I do understand the intuitive idea behind the definition of "topological space", and how it abstracts for example the notions of "neighborhood" and "near" and "far". It is not surprising that the formalism of topological spaces is useful and ubiquitous in situations involving things like R^n, subsets of R^n, manifolds, metric spaces, simplicial complexes, CW complexes, etc. 
However, when you start with algebraic objects and then get topological spaces out of them --- I find that surprising somehow because a priori there is not necessarily anything "geometric" or "topological" or "shape-y" or "neighborhood-y" going on.

Edit 2: Somebody has voted to close, saying this is "not a real question". I apologize for my imprecision and vagueness, but I still think this is a real question, for which real (mathematical) answers can conceivably exist.
For example, I'm hoping that maybe there is a theorem along the lines of something like: 

Given an algebraic object A satisfying blah, define Spec(A) to be the set of blah-blahs of A such that blah-blah-blah. There is a natural topology on Spec(A), defined by [something]. When A is a commutative ring, this agrees with the Zariski topology on the prime spectrum. When A is a commutative C^* algebra, this agrees with the [is there a name?] topology on the Gelfand spectrum. When A is a Boolean algebra... When A is a commutative Banach ring... etc.

Of course, such a theorem, if such a theorem exists at all, would also need a definition of 'algebraic object'.

REPLY [3 votes]: I know this is very late, I just happened to run into this question. 
I think Von Neumann might answer this question like this (as he once has): "One does not understand anything in mathematics one simply gets used to it." Then he may add, we have a natural functor from the category of topological spaces to the category of rings, it is an interesting natural phenomenon that this functor is strictly invertible if we restrict to suitable subcategories of the category of top spaces, and of the category of rings. This is Gelfand-Naimark.
One does not "understand" phenomena anymore then one understands why the universe exists.
By the way a topological space is also naturally an algebraic object, we can take its partially ordered set of open subsets. This often completely determines the space: http://math.nie.edu.sg/dszhao/Research%20papers/conference%20proceeding/posetmodels.pdf
(Although I didn't really read this.)