Pointless Topology

Starting with the motivating example of Stone’s representation theorem that allows one to represent Boolean algebras as subalgebras of the poweralgebra of a sufficiently large set, we ask the question of whether it is possible to generalize this to a relationship between lattice theory and topology. This can be done by considering special lattices called locales, which are, in a sense, a suitable algebraic model for a topological space. Every topological space is a locale and we can assign to each locale a topological space, which one can consider as the set of ”points” of that lattice. We will make this precise in the form of an adjunction between the category of topological spaces and the category of locales, which is the main point of this paper.