## § Conversation with Olaf Klinke

do you have reading you'd recommend to gain your viewpoint of computation-as-topology-as-computation?
I am a topologist, a domain theorist to be more precise. I had the privilege to meet many founders of this relatively young field of mathematics. Domain theory is a denotational semantics (there are others) of lambda calculus. For reading, there is the old testament and the new testament, as I call it. The old testament is "A compendium of continuous lattices" ISBN 3-540-10111-X ISBN 0-387-10111-X The new testament is "Continuous lattices and domains" ISBN 0-521-80338-1 Domain theory makes sense once one stops disregarding bottom | , or undefined. Think of domains as triangular objects, e.g. the type Bool
False   True
\   /
_|_

How does one compute a real number? Think of a horizontal real interval, and draw a triangle underneath:
---[---]---
\   \ /   /
\   .   /
\     /
\   /
_|_

Every point in the triangle represents a closed interval. At the top is a copy of the real numbers in the form of singleton intervals, the "total" elements. Every point underneath is a proper interval, representing everything reachable by "fanning out" upward from that point. This may be a model for e.g. interval arithmetic, where computing a more precise result means moving up in the triangular domain of intervals. Directed suprema (results of recursive computations) in this domain are nested intersections of intervals. Existence of these directed suprema is equivalent to the uniqueness and non-emptyness of the nested intersections, which again is guaranteed by the two topological properties "Hausfdorff" and "compactness" of the closed real interval. A treasure trove of smart little Haskell programs is