On finest unitary extensions of topological monoids (Q2515758)

Let \(\mathcal{X} \) be a Hausdorff topological monoid. A unitary Cauchy filter is a generalization of the notion of a fundamental sequence of reals. The underlying set \(X\) of the monoid \(\mathcal{X} \) endowed with the family of unitary Cauchy filters forms a Cauchy space which is called a unitary Cauchy space over \(\mathcal{X} \). In this paper, the author studies the Wyler completion of the unitary Cauchy space. It is the finest unitary extension of \(X\). He has proved that the Wyler completion \(\bar{X}\) of the unitary Cauchy space over \(\mathcal{X} \) is a \(T_2\)-topological space. Moreover, he showes that the function \(\mathcal{X} \to \bar{X}\) generates a covariant functor from the category of topological monoids and their homomorphisms into the category of \(T_2\)-topological spaces. If \(\mathcal{X} \) is commutative, then \(\bar{X}\) is an abstract monoid and a canonical homeomorphic embedding \(i\) of \(X\) endowed with its unitary topology into \(\bar{X}\) is an algebraic embedding.