Tame topology

Alexander Grothendieck suggested creating a new branch of topology, called by him ``topologie modérée''. In the paper ``On Grothendieck's tame topology" by N. A'Campo, L. Ji and A. Papadopoulos (Handbook of Teichmüller Theory, Volume VI. IRMA Lectures in Mathematics and Theoretical Physics Vol. 27 (2016), pp. 521-533) the authors conclude that no such tame topology has been developed on the 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 getting several equivalences of categories of spaces with categories of lattices possible. We show several variants of Stone Duality and Esakia Duality for categories of small spaces 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. In particular, 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, it is better to look at them as topological spaces with additional structure. 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 in the case of developing o-minimal homotopy theory.