§ 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.