I was browsing through Wikipedia today when I came across the definition of pretopological space. The notion seemed very exotic until I thought of a family of examples, which I’m christening ennui spaces.
A pretopological space prescribes for each point the set of (not-necessarily open) neighborhoods of that point. The set of neighborhoods of a given point are required to satisfy some natural axioms, but neighborhoods of one point can be completely unrelated to neighborhoods of another point. A sequence in a pretopological space converges if for any neighborhood, the sequence eventually enters that neighborhood and never leaves it again. A topological space can be turned into a pretopological space by taking as the set of neighborhoods of a point to be all sets that contain an open set that contain that point. You can try to reverse the process by borrowing the characterization of the closure of a set in terms of sequences (or nets), but usually the topological space you construct will have a coarser notion of convergence than the pretopological space.
An ennui space has the same underlying set as a metric space. A neighborhood of a point is any set that contains the unit ball around that point. A sequence in an ennui space converges to a point if is guaranteed to be eventually within one unit of the point. The mental image I have is that the sequence gets close to its destination, but then gets bored. If you try to construct a topology out of this space, you get the indiscrete topology, where all sequences converge to all points. Essentially, all information about the convergence properties of the ennui space are lost.
A practical example of an ennui space would be your computer whenever it simulates a convergent sequence of operations, such as numerical integration or Newton’s method. The computer gets within machine precision of the correct answer, and then stops to light up a Gauloise and discuss L’ÃŠtre et le nÃ©ant in a cafe.