## ยง Why pointless topology is powerful

- Key idea of pointless topology: topology manipulates open sets and their lattice. Forget the set, simply manipulate lattices!
- When can a lattice be written in terms of sets?
- Birkhoff representation theorem: Lattice is distributive iff isomorphic to a lattice of subsets of join-irreducible elements.
- Hence, if we take
*non distributive lattices *, we have geometry (locale) which *has no incarnation * as subsets! - Yay, extra power.