On unitary Cauchy filters on topological monoids

Abstract

For Hausdorff topological monoids, the concept of a unitary Cauchy net is a generalization of the concept of a fundamental sequence of reals. We consider properties and applications of such nets and of corresponding filters and prove, in particular, that the underlying set of a given monoid, endowed with the family of such filters, forms a Cauchy space whose convergence structure defines a uniform topology. A commutative monoid endowed with the corresponding uniformity is uniform. A distant purpose of the paper is to transfer the classical concepts of a completeness and of a completion into the theory of topological monoids.

On finest unitary extensions of topological monoids

“…By Corollary 2.4 from [1], for each C-filter F, there exists a unique equivalent C-filter which properly contains no other C-filter. It is equal to the intersection of all C-filters from the equivalence class of F and is denoted by F lst .…”

“…By Proposition 1.7 from [1], ≥ is a quasi-order relation, and ≈ is an equivalence relation on the set of Cfilters. Similar relations for left (right) C-filters are denoted by ≥ L , ≥ R , ≈ L , and ≈ R .…”

“…It was proved in [1] that the underlying set X of this monoid X endowed with the family of such filters forms a Cauchy space called a unitary Cauchy space over X. Its convergence structure defines a T 3 1 2 -topology on X which is also said to be unitary.…”

“…The most important of them is perhaps the Wyler completion possessing the universal property over other completions. The proved in [1] existence of the least with respect to inclusion C-filters allows us to simplify its construction and proofs of its properties. In particular, we prove that the Wyler completionX of the unitary Cauchy space over X is a T 2 -topological space, and the function X →X generates a covariant functor from the category of topological monoids and their homomorphisms into the category of T 2 -topological spaces.…”

“…For an arbitrary Hausdorff topological monoid, the concept of a unitary Cauchy filter (for brevity, we call them C-filters) which was defined in [1], generalizes the notion of a fundamental sequence of reals. It was proved in [1] that the underlying set X of this monoid X endowed with the family of such filters forms a Cauchy space called a unitary Cauchy space over X.…”