Topology without points (or sets, of which points are elements). Strictly speaking, a "topology" is merely a collection of objects (called "open sets") that satisfy certain conditions. Replacing the concept of subsets of "the space" with any lattice (poset) that is closed to finite meets (^) and any joins (v), we can specify a pointless topology:
  1. The maximum and minimum are open thingies.
  2. The set of open thingies is closed to finite meets.
  3. The set of open thingies is closed to any joins.

This lets us define a topology even without identifying the points of the space! It lets you do some interesting category theory things, since it abstracts away the "specific" character of topology, leaving only the "generic" character.

Used (so I am told) in some advanced algebra, especially algebraic number theory.