In the context of metric (and in general first-countable) topologies, it's reasonably clear what a closed set is: a set $F$ is closed if and only if every convergent sequence of points in $F$ converges to a point also in $F$. This naturally generalises to the definition of a closed set in an arbitrary topological space using the concept of limit points… but limit points are defined in terms of open sets, which are, to me, somewhat more mysterious than closed sets.
I was once told that ultrafilters axiomatise the concept of (sets of) big sets. I'm hoping here to find a similar conceptual picture of topologies defined by systems of open sets, preferably without reference to closed sets. One such explanation I've seen is that open set axiomatise the concept of nearness, which, I suppose, is fair enough at least for metric topologies. Indeed, if something is "happening" at points near $x \in X$, then it's usually the case that every open neighbourhood $U \subseteq X$ of $x$ contains an open subset $V \subseteq U$, $x \in V$, such that the something "happens" at all points in $V$. But what about non-metrisable topologies, particularly the coarse ones where there are no "small" open sets?
Consider, for example, the Zariski topology on affine $n$-space $\mathbb{A}^n$. $\mathbb{A}^n$ is irreducible, so every non-empty open set is dense. It seems reasonable to interpret this to mean that every non-empty open set is "large". Indeed, if we work over the complex numbers, then in the usual metric geometry, Zariski-open sets are unbounded and have full measure, so are very large indeed, so I don't think it's fair to say that open sets are capturing the notion of nearness here.
I'm also curious about the history of point set topology. When were the axioms first written down? What were the first "non-geometric" examples of topological spaces — "non-geometric" here referring to either non-metrisable topologies or topologies on sets other than sets of points of some intuitively geometric object — and were they part of the motivations for creating point set topology?
Best Answer
Edit, 7/10/11: The idea below was recently also discussed in this blog post by Michael O'Connor.
This was thoroughly discussed on MathOverflow. The answers that I found most convincing can be summarized as follows: open sets axiomatize the notion of a semidecidable property.
That is, open sets axiomatize the notion of a condition whose truth can be verified in finite time (but whose falsehood cannot necessarily be verified in finite time). A continuous function $f : A \to B$ between two spaces is a function such that the preimage of any semidecidable subset is semidecidable, hence is a computable function in the sense that a decision procedure to verify whether $f(a) \in U \subset B$ in finite time gives a decision procedure to verify whether $a \in V \subset A$ in finite time.
To really make sense of what I just said above you should think of $A$ as the set of possible conditions of some system, $f$ as a measurement of some property of the system, and $B$ as the set of possible values of the property you're measuring. Then $f$ is computable precisely when information about $f(a)$ allows us to deduce information about $a$. In some sense it is the central premise of the scientific method that this is possible.
Note that the above description really brings out the special role of the Sierpinski space $\mathbb{S}$. Indeed, a subset of a topological space $X$ is open precisely when the indicator function $X \to \mathbb{S}$ is continuous.
Onto the axioms:
Finally, note that it is intuitively possible to verify whether a point in a metric space lies in an open ball in finite time by showing that it lies in an even smaller open ball (which can be done using finite-precision computations), but it is not necessarily possible to do the same for a closed ball because the point may lie on the boundary and we cannot make arbitrarily precise measurements in finite time.
In the particular case of the Zariski topology, it is always possible to verify in finite time whether a polynomial is nonzero at a point by computing with sufficient precision, but without additional information it is not necessarily possible to verify in finite time whether a polynomial is zero at a point.