§ Pointless topology: Frames
- A frame is a lattice with arbitrary joins, finite meets, with distributive law: .
- A map of frames is a lattice map between frames.
- A category of locales is the opposite category of frames.
§ Thm: Any locale has a smallest dense sublocale
- For example, has .
§ Sober spaces
- A space is sober iff every irreducible closed subset is the closure of a single point.
- A sober space is one whose lattice of open subsets determine the topology of the space.
- A space is sober iff for every topological embedding that adds more points to , if the inverse image map is an isomorphism, then is a homeomorphism. Source: martin escardo twitter This means we can't add more points to without changing its topology. it has as many points as it could.
- Equivalently: Every complete prime filter of open sets is the open nbhd filter of a unique point.
- is a completely prime filter iff (1) is closed under all finite intersections (including empty), (2) if the union of some family is in , then some is already in (prime).
- This tries to specify a point by open sets.
- Joke: A sober space is one where what you see is there, and you don't see double. What you see is there: every completely prime filter is the nbhd of some point. You don't see double: the pt is unique.