Non-Hausdorff Topology and Domain Theory

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We pursue last month’s post, written with Zhenchao Lyu, and we describe Gierz’s example of a compact semilattice — namely, a compact Hausdorff topological semilattice — which does not have small semilattices. It is a bit simpler than Jimmie Lawson’s counterexample, which was already described on this blog. Read the full post.

There is a well-known formula in domain theory, which, given a monotonic map f from a basis B of a continuous poset X to a dcpo Y, produces the largest continuous map f’ defined on the whole of X and below f on B. This is Scott’s formula: f’(x) ≝ sup_{b ∈ B, b ≪ x} f(b). Generalizing this to the setting where X is a c-space, I will explain when exactly f’ coincides with f on B, hence when f’ is a continuous extension of f. Also, using a motivation from the theory of continuous valuations, I will describe conditions under which any algebraic laws satisfied by f transfer to f’. Read the full post.

Recently, Achim Jung sent me a message from Jimmie Lawson, and suggested that I might be interested in posting the information on this blog. The red book [1] is a precious source of information on domain theory, and if you are interested in knowing how the material there came to be discovered, Continue reading →