Stone duality for Kolmogorov locally small spaces

Abstract: We prove three new versions of Stone Duality. The main version is the following: the category of Kolmogorov locally small spaces and bounded continuous mappings is equivalent to the category of spectral spaces with decent lumps and with bornologies in the lattices of compact (not necessarily Hausdorff) open sets as objects and spectral mappings respecting the decent lumps and satisfying a boundedness condition as morphisms as well as it is dually equivalent to the category of bounded distributive lattices with bornologies and with decent lumps of prime filters as objects and homomorphisms of bounded lattices respecting those decent lumps and satisfying a domination condition as morphisms. Some theory of strongly locally spectral spaces is developed.