Tame Topology

Abstract

Alexander Grothendieck suggested creating a new branch of topology, called “topologie modérée”. In a paper by N. A’Campo, L. Ji, and A. Papadopoulos the authors conclude that no such tame topology has been developed at a purely topological level. We see our theory of sets with distinguished families of subsets, which we call smopologies, as realising Grothendieck’s idea and the demands of the mentioned paper. Dropping the requirement of stability under infinite unions makes it possible to obtain several equivalences of categories of spaces with categories of lattices. We show several variations in Stone duality and Esakia duality for categories of small or locally small spaces and some subclasses of strictly continuous (or bouned continuous) mappings. Such equivalences are better than the spectral reflector functor for usual topological spaces. Some spectralifications of Kolmogorov locally small spaces can be obtained by Stone duality. Small spaces or locally small spaces seem to be generalised topological spaces. However, looking at them as topological spaces with an additional structure is better. The language of smopologies and bounded continuous mappings simplifies the language of certain Grothendieck sites and permits us to glue together infinite families of definable sets in structures with topologies, which was important when developing the o-minimal homotopy theory.

Keywords

• tame topology
• Stone duality
• Esakia duality
• spectralification
• Grothendieck site
• o-minimal structure