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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?

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'.
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Why should algebraic objects have naturally associated topological spaces? (Formerly: What is a topological space?space?)

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: 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? If 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.
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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 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: I should clarify a bit. One question that I meant to and should have asked isLet 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? If these things are not coincidences, then what's the explanation?

Of course I do understand the intuitive idea behind the definition of "topological spacespace", and how it abstracts for example the notion notions of "neighborhood". neighborhood" and "near" and "far". It is not surprising to me that the formalism of topological spaces will arise is useful and ubiquitous in situations that are "geometric" or "topological" or "shape-y" or "neighborhood-y": for example involving things like R^n, subsets of R^n, manifolds, metric spaces, simplicial complexes, CW complexes, metric spacesetc.

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.
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