Spectral Spaces

Max Dickmann has been a researcher at the Centre National de la Recherche Scientifique (CNRS), Paris, since 1974, Directeur de Recherche since 1988 and emeritus since 2007. His research interests include the applications of spectral spaces to real algebraic geometry, quadratic forms, and related topics. Niels Schwartz is Professor of Mathematics at the Universität Passau, Germany, retired since 2016. Many of his publications are concerned with, or use, spectral spaces in essential ways. In particular, he has used spectral spaces to introduce the notion of real closed rings, an important topic in real algebra and geometry. Marcus Tressl is a mathematician working in the School of Mathematics at the University of Manchester. His research interests include model theory, ordered algebraic structures, ring theory, differential algebra, and non-Hausdorff topology.

Outline of the history of spectral spaces; 1. Spectral spaces and spectral maps; 2. Basic constructions; 3. Stone duality; 4. Subsets of spectral spaces; 5. Properties of spectral maps; 6. Quotient constructions; 7. Scott topology and coarse lower topology; 8. Special classes of spectral spaces; 9. Localic spaces; 10. Colimits in Spec; 11. Relations of Spec with other categories; 12. The Zariski spectrum; 13. The real spectrum; 14. Spectral spaces via model theory; Appendix. The poset zoo; References; Index of categories and functors; Index of examples; Symbol index; Subject index.