Stone duality for Kolmogorov locally small spaces

TL;DR

This paper establishes three new versions of Stone Duality connecting categories of locally small spaces, spectral spaces, and bounded distributive lattices, expanding duality theory in topology and lattice theory.

Contribution

It introduces novel duality theorems linking Kolmogorov locally small spaces with spectral spaces and bounded lattices, including the development of strongly locally spectral spaces.

Findings

Categories of locally small spaces and spectral spaces are equivalent.

Spectral spaces with decent lumps and bornologies correspond to bounded distributive lattices.

New duality theorems extend classical Stone Duality to broader classes of spaces and lattices.

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.