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'.
Best Answer
I will take the question at face value, but not in the sense of justifying the definition.
A topological space is a convenient way of encoding, or perhaps better, organising, certain types of information. (Vague but true! I will give some instances. the data is sometimes `spatial' but more often than not, is not.)
Perhaps we should not think of spaces as 'god given' merely 'convenient', and there are variants that are more appropriate in various contexts.
A related question, coming from an old Shape Theorist (myself) is : when someone starts a theorem with 'Given a space $X$...', how is the space 'given'? As an algebraic topologist I sometimes need to use CW-complexes, but face the inconvenience that if I could give the CW structure precisely I could probably write down an algebraic model for its homotopy type precisely, and vice versa, so a good model is exactly the same as the one I started with. I hoped for more insight into what the space 'was' from my modelling. Giving the space is the end of the process, not the beginning. Strange. A space is a pseudo-visual way of thinking about 'data', which encodes important features, or at leastsome features that we can analyse, partially.
If someone gives me a compact subspace of $\mathbb{R}^n$, perhaps using some equations and inequalities, can I work out algebraic invariants of its homotopy type, rather than just its weak homotopy type? The answer will usually be no. Yet important properties of $C^*$ algebras on such a space, can sometimes be related to algebraic topological invariants of the homotopy type.
Spaces can arise as ways of encoding actual data as in topological data analysis, where there is a 'cloud' of data points and the practitioner is supposed to say something about the underlying space from which the data comes. There are finitely many data points, but no open sets given, they are for the data analyst to 'divine'.
Not all spatial data is conveniently modelled by spaces as such and directed spaces of various types have been proposed as models for changing data. Models for space-time are like this, but also models for concurrent systems.
Looking at finite topological spaces is again useful for encoding finite data (and I have rarely seen infinite amounts of data). For instance, relations between finite sets of data can be and are modelled in this way. Finite spaces give all homotopy types realisable by finite simplicial complexes. Finite spaces can be given precisely (provided they are not too big!) How do invariants of finite spaces appear in their structure? (Note the problem of infinite intersections does not arise here!!!)
At the other extreme, do we need points? Are locales not cleaner beasties and they can arise in lots of algebraic situations, again encoding algebraic information. Is a locale a space?
I repeat topological spaces are convenient, and in the examples you cite from algebraic geometry they happen to fit for good algebraic reasons. In other contexts they don't. Any Grothendieck topos looks like sheaves on a space, but the space involved will not usually be at all `nice' in the algebraic topological sense, so we use the topos and pretend it is a space, more or less.