§ Pointless topology: Frames

• A frame is a lattice with arbitrary joins, finite meets, with distributive law: $A \cap \cup_i B_i = \cup_i A \cap B_i$.
• 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, $\mathbb R$ has $\mathbb Q$.

§ 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 $X$ is sober iff for every topological embedding $f: X \to X'$ that adds more points to $X$, if the inverse image map $f: O(X') \to O(X)$ is an isomorphism, then $f$ is a homeomorphism. Source: martin escardo twitter This means we can't add more points to $X$ 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.
• $F \subseteq O(X)$ is a completely prime filter iff (1) $F$ is closed under all finite intersections (including empty), (2) if the union of some family $O_i$ is in $F$, then some $O$ is already in $F$ (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.