Topology can be defined directly, without open sets, as the study of "metric spaces without the metric", i.e., modulo metric deformations or homeomorphism. This matches reasonably well the intuition of a qualitative geometry, insensitive to stretching and bending.
One can then prove that the structure of open sets is a complete invariant (since we start from metrizable spaces), and one can observe, with some experience, that reasoning about the topology of metric spaces (i.e., proofs of properties that are invariant under deformation or homeomorphism) can be formulated directly in terms of this invariant. In other words, not only the topologically invariant maps but the constructions of those maps descend to the category of topological spaces in terms of open sets. This means that we can work natively in manifestly topologically invariant terms provided that the invariant thing --- the structure of open sets --- is taken as the object of study. This is a rare case of a total or near-total success of the Erlangen program, where thinking in terms of that which is invariant really suffices for the original purposes of the subject.
(I say near-total, but don't know of any example where topologically invariant properties of a metric spaces are most easily proved using one or more metrics.)
Once topology is set up in terms of open sets one can look at examples beyond the motivating intuition, such as Zariski topology, the long line or pathological spaces. As far as those extensions start to challenge the adequacy of the open-set formalism it is because they are based on phenomena different from the stretching and bending ideas abstracted from picturesque low-dimensional situations.