Profinite Topological Spaces

The finite-dimensional Galois theorems of the previous chapters admit generalizations to arbitrary dimensions. This requires superposing topological structures on the algebraic ones. These topological aspects do not appear explicitly in the finite-dimensional cases, just because the topologies involved are then discrete. The aim of the present chapter is to develop the useful topological ingredients in view of proving infinite-dimensional Galois theorems. They will be obtained by a limit process from the finite discrete case. A topological space is profinite when it is compact Hausdorff and its topology admits a basis of closed open subsets, or equivalently, when it is a limit of finite discrete spaces. The Stone duality exhibits a contravariant equivalence of categories between the category of profinite spaces and that of Boolean algebras. This link will make it possible to combine algebraic and topological aspects in the infinite-dimensional Galois theory of fields, but also in the Galois theory of rings.