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:

- The maximum and minimum are open thingies.
- The set of open thingies is closed to finite meets.
- 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.