D. C. Kent

Approach Spaces, Limit Tower Spaces, and Probabilistic Convergence Spaces

The category LTS of limit tower spaces is defined and shown to be isomorphic to the category CAP of convergence approach spaces. The full subcategory of LTS determined by the objects satisfying a diagonal axiom due to Cook and Fischer is shown to be isomorphic to the category AP of approach spaces. A family of isomorphisms is also obtained between LTS and certain full subcategories of the category PCS of probabilistic convergence spaces.

On Convergence Approach Spaces

The extension of two axioms due to Cook and Fischer to the category CAP of convergence approach spaces leads to the study of non-Archimedean approach spaces as well as two versions of regularity appropriate to CAP and related categories.

Completion functors for Cauchy spaces

Completion functors are constructed on various categories of Cauchy spaces by forming the composition of Wylers completion functor with suitable modification functors.

On the Nachbin compactification of products of totally ordered spaces

Necessary and sufficient conditions are given for β 0 ( X × Y ) = β 0 X × β 0 Y , where X and Y are totally ordered spaces and β 0 X denotes the Nachbin (or Stone-Cech ordered) compactification of X .

A note on ordered planes

Necessary and sufficient conditions are given for the equivalence of the Nachbin and Wallman-ordered compactification of an ordered plane.

Separation properties of the Wallman ordered compactification

The Wallman ordered compactification ω0X of a topological ordered space X is T2-ordered (and hence equivalent to the Stone-Cech ordered compactification) iff X is a T4-ordered c-space. In particular, these two ordered compactifications are equivalent when X is n dimensional Euclidean space iff n≤2. When X is a c-space, ω0X is T1-ordered; we give conditions on X under which the converse statement is also true. We also find conditions on X which are necessary and sufficient for ω0X to be T2. Several examples provide further insight into the separation properties of ω0X.

A new ordered compactification

A new Wallman-type ordered compactification γ ∘ X is constructed using maximal C Z -filters (which have filter bases obtained from increasing and decreasing zero sets) as the underlying set. A necessary and sufficient condition is given for γ ∘ X to coincide with the Nachbin compactification β ∘ X ; in particular γ ∘ X = β ∘ X whenever X has the discrete order. The Wallman ordered compactification ω ∘ X equals γ ∘ X whenever X is a subspace of R n . It is shown that γ ∘ X is always T 1 , but can fail to be T 1 -ordered or T 2 .

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 that T2 and T3 modifications 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 have T3 completions.

A completion functor for Cauchy groups

A completion functor is constructed on the category of completely normal Cauchy groups and Cauchy-continuous homomorphlsms. A competlon functor is also obtained for a corresponding category of convergence groups.

p-regular Cauchy completions

This paper gives a further development of p-regular completion theory, including a study of p-regular Reed completions, the role of diagonal axioms, and the relationship between p-regular and p-topological completions.