Quantale-valued Cauchy tower spaces and completeness

Generalizing the concept of a probabilistic Cauchy space, we introduce quantale-valued Cauchy tower spaces. These spaces encompass quantale-valued metric spaces, quantale-valued uniform (convergence) tower spaces and quantale-valued convergence tower groups. For special choices of the quantale, classical and probabilistic metric spaces are covered and probabilistic and approach Cauchy spaces arise. We also study completeness and completion in this setting and establish a connection to the Cauchy completeness of a quantale-valued metric space.

On finest unitary extensions of topological monoids

We prove that the Wyler completion of the unitary Cauchy space on a given Hausdorff topological 5 monoid consisting of the underlying set of this monoid and of the family of unitary Cauchy filters on it, is a T2-topological space and, in the commutative case, an abstract monoid containing the initial one.

On unitary Cauchy filters on topological monoids

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.

Completion of a Cauchy space without theT2-restriction on the space

A completion of a Cauchy space is obtained without theT2restriction on the space. This completion enjoys the universal property as well. The class of all Cauchy spaces with a special class of morphisms calleds-maps form a subcategoryCHY' ofCHY. A completion functor is defined for this subcategory. The completion subcategory ofCHY' turns out to be a bireflective subcategory ofCHY'. This theory is applied to obtain a characterization of Cauchy spaces which allow regular completion.

p-topological Cauchy completions

The duality between “regular” and “topological” as convergence space properties extends in a natural way to the more general properties “p-regular” and “p-topological.” Since earlier papers have investigated regular,p-regular, and topological Cauchy completions, we hereby initiate a study ofp-topological Cauchy completions. Ap-topological Cauchy space has ap-topological completion if and only if it is “cushioned,” meaning that each equivalence class of nonconvergent Cauchy filters contains a smallest filter. For a Cauchy space allowing ap-topological completion, it is shown that a certain class of Reed completions preserve thep-topological property, including the Wyler and Kowalsky completions, which are, respectively, the finest and the coarsestp-topological completions. However, not allp-topological completions are Reed completions. Several extension theorems forp-topological completions are obtained. The most interesting of these states that any Cauchy-continuous map between Cauchy spaces allowingp-topological andp′-topological completions, respectively, can always be extended to aθ-continuous map between anyp-topological completion of the first space and anyp′-topological completion of the second.

Ordered Cauchy spaces

This paper is concerned with the notion of “ordered Cauchy space” which is given a simple internal characterization in Section 2. It gives a discription of the category of ordered Cauchy spaces which have ordered completions, and a construction of the “fine completion functor” on this category. Sections 4 through 6 deals with certain classes of ordered Cauchy spaces which have ordered completions; examples are given which show that the fine completion does not preserve such properties as uniformizability, regularity, or total boundedness. From these results, it is evident that a further study of ordered Cauchy completions is needed.

On the Regularity of the Kowalsky Completion

Cauchy spaces were introduced by Kowalsky in 1954 [9]. In that paper a first completion method for these spaces was given. In 1968 Keller [5] has shown that the Cauchy space axioms characterize the collections of Cauchy filters of uniform convergence spaces in the sense of [1]. Moreover in the completion theory of uniform convergence spaces the associated Cauchy structures play an essential role [12]. This fact explains why in the past ten years in the theory of Cauchy spaces, much attention has been given to the study of completions.

The regularity series of a Cauchy space

This study extends the notion of regularity series from convergence spaces to Cauchy spaces, and includes an investigation of related topics such as thatT2andT3modifications of a Cauchy space and their behavior relative to certain types of quotient maps. These concepts are applied to obtain a new characterization of Cauchy spaces which haveT3completions.

Regular completions of Cauchy spaces via function algebras

A regular completion with the universal property is obtained for each member of a certain class of Cauchy spaces by embedding the Cauchy space in a complete function algebra with the continuous convergence structure.