In this answer I will combine ideas of sigfpe's answer, sigfpe's blog, the book by Vickers, Kevin's questions and Neel's answers adding nothing really new until the last four paragraphs, in which I'll attempt to settle things about the open vs. closed ruler affair.
DISCLAIMER: I see that some of us are answering a question that is complementary of the original, since we are trying to motivate the structure of a topology, instead of adressing the question of which of the many equivalent ways to define a topology should be used, which is what the question literally asks for. In the topology course that I attended, it was given to us in the first class as an exercise to prove that a topology can be defined by its open sets, its neighbourhoods, its closure operator or its interior operator. We later saw that it can also be stated in terms of convergence of nets. Having made clear these equivalence of languages, its okay that anyone chooses for each exposition the language that seems more convinient without further discussion. However, I will mantain my non-answer since many readers have found the non-question interesting.
Imagine there's a set X of things that have certain properties. For each subset of S there is the property of belonging to S, and in fact each property is the property of belonging to an adequate S. Also, there are ways to prove that things have properties.
Let T be the family of properties with the following trait: whenever a thing has the property, you can prove it. Let's call this properties affirmative (following Vickers).
For example, if you are a merchant, your products may have many properties but you only want to advertise exactly those properties that you can show. Or if you are a physicist, you may want to talk about properties that you can make evident by experiment. Or if you predicate mathematical properties about abstract objects, you may want to talk about things that you can prove.
It is clear that if an arbitrary family of properties is affirmative, the property of having at least one of the properties (think about the disjunction of the properties, or the union of the sets that satisfy them) is affirmative: if a thing has at least one of the properties, you can prove that it has at least one of the properties by proving that property that it has.
It is also clear that if there is a finite family of affirmative properties, the property of having all of them is affirmative. If a thing has all the properties, you produce proofs for each, one after the other (assuming that a finite concatenation of proofs is a proof).
For example, if we sell batteries, the property P(x)="x is rechargeable" can be proved by putting x in a charger until it is recharged, but the property Q(x)="x is ever-lasting" can't be proved. It's easy to see that the negation of an affirmative property is not necessarily an affirmative property.
Let's say that the open sets are the sets whose characteristic property is affirmative. We see that the family T of open sets satisfies the axioms of a topology on X. Let's confuse each property with the set of things that satisfy it (and open with affirmative, union with disjunction, etc.).
Interior, neighbourhood and closure: If a property P is not affirmative, we can derive an affirmative property in a canonical way: let Q(x)="x certainly satisfies P". That is, a thing will have the property Q if it can be proved that it has the property P. It is clear that Q is affirmative and implies P. Also, Q is the union of the open sets contained in P. Then, it is the interior of P, which is the set of points for which Q is a neighbourhood. A neighbourhood of a point x is a set such that it can be proved that x belongs to it. The closure of P is the set of things that can't be proved not to satisfy P.
Axioms of separation: If T is not T0, there are x, y that can't be distinguished by proofs and if it is not T1, there are x, y such that x can't be distinguished from y (we can think that they are apparently identical batteries, but x is built in such a way that it will never overheat. So if it overheats, then it's y, but if it doesn't, you can't tell).
Base of a topology: Consider a family of experiments performable over a set X of objects. For each experiment E we know a set S of objects of X over which it yields a positive result (nothing is assumed about the outcome over objects that do not belong to S). If you consider the properties that can be proved by a finite sequence of experiments, the sets S are affirmative and the topology generated by them is the family of all the affirmative properties.
Compactness: I don't know how to interpret it, but I think that some people know, and it would be nice if they posted it. (Searchable spaces?)
Measurements: A measurement in a set X is an experiment that can be performed on each element of X returning a result from a finite set of possible ones. It may be a function or not (it is not a function if there is at least one element for which the result is variable). The experiment is rendered useful if we know for each possible result r a set T_r of elements for which the experiment certainly renders r and/or a set F_r for which it certainly doesn't, so let's add this information to the definition of measurement. An example is the measurement of a length with a ruler. If the length corresponds exactly with a mark on the ruler, the experimenter will see it and inform it. If the length fits almost exactly, the experimenter may think that it fits a mark or may see that it doesn't. If the length clearly doesn't fit any mark (because he can see that it lies between two marks, or because the length is out of range), he will inform it. It is sufficient to study measurements that have only a positive outcome and a negative outcome, a set T for which the outcome is certainly positive and a set F for which the outcome is certainly negative.
Imprecise measurements on a metric space: If X is a metric space, we say that a measurement in X is imprecise if there isn't a sequence x_n contained in F that converges to a point x contained in T. Suppose that there is a set of imprecise measurements available to be performed on the metric space. Suppose that, at least, for each x in X we have experiments that reveal its identity with arbitrary precision, that is, for each e>0 there is an experiment that, when applied to a point y, yields positive if y=x and doesn't yield positive if d(y,x)>e. Combining these experiments we are allowed to prove things. What are the affirmative sets generated by this method of proof? Let S be a subset of X. If x is in the (metric) interior of S, then there is a ball of some radius e>0 centered at x and contained in S. It is easy to find an experiment that proves that x belongs to S. If x is in S but not in the interior (i.e, it is in the boundary), we don't have a procedure to prove that x is in S, since it would involve precise measurement. Therefore, the affirmative sets are those that coincide with its metric interior. So, the imprecise measurements of arbitrary precision induce the metric topology.
Experimental sciences: In an experimental science, you make a model that consists of a set of things that could conceivably happen, and then make a theory that states that the things that actually happen are the ones that have certain properties. Not all statements of this kind are completely meaningful, but only the refutative ones, that is, those that can be proved wrong if they are wrong. A statements is refutative iff its negation is affirmative. By applying the closure operator to a non refutative statement we obtain a statement that retains the same meaning of the original, and doesn't make any unmeaningful claim.
An example from classical physics: Assume that the space-time W is the product of Euclidean space and an affine real line (time). It can be given the structure of a four-dimensional real normed space. Newton's first law of motion states that all the events of the trajectory of a free particle are collinear in space-time. To prove it false, we must find a free particle that incides in three non-collinear events. This is an open condition predicated over the space W^3 of 3-uples of events, since a small perturbation of a counterexample is also a counterexample. Assuming that imprecise measurements of arbitrary precision can be made, it is an affirmative property. I think that classical physicists, by assuming that these kind of measurements can be done, give exact laws like Newton's an affirmative set of situations in which the law is proved false. I also suspect (but this has more philosophical/physical than mathematical sense) that the mathematical properties of space-time (i.e. that it is a normed space over an Archimedean field) are deduced from the kind of experiments that can be done on it, so there could be a vicious circle in this explanation.