Introduction to the Topology of Spectral Spaces

Emmanuel Godard (LIS)

Summary

We will present a serie of classical results about spectral spaces. Spectral spaces are non-Haussdorf topological spaces that have seen many applications in computer science. They are a tool linking algebraic structures, in a very wide sense, with geometry. In applications, it is a way to relate behaviours of programs with specifications.

A spectral space is a limit of finite $T_0$ spaces, where $T_0$ denotes the first level of topological distinguishability. We focus on topological properties of such spaces with a look at the various existing topological indistinguishability (specialization order, $T_1$, $T_2$, regular, normal spaces, ...), hinting at the underlying order theoretic / lattices structures of those spaces, that can also be defined algebraically as specter of commutative rings.

Finally we present some properties of the category of spectral spaces and spectral maps, and time permitting, the duality with the category of Distributed Lattices.