Frames · Locales · Sublocales
Topology
without points
Take a space. Throw away every point. Keep only the lattice of open sets and the way those opens meet and join. Almost nothing is lost — and a great deal becomes clearer.
This is a small gallery for that idea and for the people who built it. We begin with Bernhard Banaschewski, who spent sixty-seven years at one desk (and around the world) exploring and perfecting the ideas.
Upcoming · Orange, California
Banaschewski 100A centennial conference for Bernhard Banaschewski
11–14 January 2027 at Chapman University. Four days of talks among the people who work on frames and locales — with 8 invited speakers and a documentary premiere on the opening night.
Opens in
The move
The opens were doing all the work anyway
A topological space is a set of points together with a chosen family of open subsets. Almost every theorem in topology is a statement about those opens — how they intersect, how they cover, how they separate. The points sit underneath, quietly, mostly unused.
So take them away. What is left is a complete lattice in which finite meets distribute over arbitrary joins. That object is called a frame, and when you want to think of it geometrically rather than algebraically you call it a locale.
The surprise is not that this works. The surprise is that the pointfree category behaves better. Sublocales form a complete lattice where subspaces did not. Products do not misbehave. Tychonoff’s theorem stops needing the axiom of choice. Results that were delicate become structural.
Every space has a frame of opens. Every locale has a space of points. The two constructions are adjoint, and the adjunction becomes an equivalence exactly on the sober spaces and the spatial locales.
The locales left outside are not pathologies. They are the whole point — locales with almost no points, which nonetheless behave impeccably.
Bernhard
Banaschewski
Born in Munich in 1926. Doctorate at Hamburg under Ernst Witt. In 1955 he joined the mathematics department at McMaster University in Hamilton, Ontario — and never left, chairing it twice, holding the McKay Professorship from 1964, and continuing to publish for three decades after his retirement in 1991.
Over two hundred papers, across topology, order theory, category theory, algebra and logic. More than anyone else, he is the reason pointfree topology is a complete theory rather than a promising reformulation: he went through classical topology result by result and found what each one becomes when the points are gone.
Compactification, classified
With Christopher Mulvey he built the Stone–Čech compactification of a locale — no points, no choice. And he proved the structural fact underneath it: compactifications of a frame correspond order-isomorphically to the strong inclusions on it.
The pointfree real line
A whole book on what the real numbers look like when you refuse to build them out of points, and on the rings of continuous real functions that live over them.
Proofs that travel
Decades spent tracing exactly which theorems need the axiom of choice and which only appeared to. A proof without choice is a proof that holds in any topos, not just in sets.