Cauchy completions of nearness frames

A nearness frame is Cauchy complete if every regular Cauchy filter on the nearness frame is convergent and we show that the categoryCCNFrm of Cauchy complete nearness frames is coreflective in the categoryNFrmC of nearness frames and Cauchy homomorphisms and that the coreflection of a nearness frame is given by the strict extension associated with regular Cauchy filters on the nearness frame. Using the same completion, we show that the categoryCCSNFrm of Cauchy complete strong nearness frames is coreflective in the categorySNFrm of strong nearness frames and uniform homomorphisms.     

Remarks on some coreflexive subcategories of nearness frames

Summary We introduce a separable nearness frame and construct the separable coreflection of a nearness frame. Certain aspects of the separable, the totally bounded, the compact regular (or Samuel compactification) and uniform coreflections of a nearness frame are considered.     

Strong Nearness Frames Having Enough Cauchy Filters

We prove results establishing sufficient conditions for the sum of two nearness frames to have enough Cauchy filters. From these results and the fact that, in the category of strong nearness frames having enough Cauchy filters and uniform frame maps, complete spatial frames form a coreflective subcategory, follow a variety of results where the open-sets contravariant functor from topological spaces to frames transforms products into sums and inverse limits into direct limits.     

Some notes on connectedness in nearness frames

Abstract

We study the uniform connection properties of uniform local connectedness, a weaker variant of the latter, and a certain property S in the context of nearness frames. We show that the uniformly locally connected nearness frames form a reflective subcategory of the category of nearness frames whose underlying frame is locally connected. Amongst other results we show that these uniform connection properties are conserved and reflected by perfect nearness extensions which are uniformly regular.     

The Stone-Čech compactification of a partial frame via ideals and cozero elements

A partial frame is a meet-semilattice in which certain designated subsets are required to have joins, and finite meets distribute over these. The designated subsets are specified by means of a so-called selection function, denoted by S ; these partial frames are called S-frames.

We construct free frames over S-frames using appropriate ideals, called S-ideals. Taking S-ideals gives a functor from S-frames to frames. Coupled with the functor from frames to S-frames that takes S-Lindelöf elements, it provides a category equivalence between S-frames and a non-full subcategory of frames. In the setting of complete regularity, we provide the functor taking S-cozero elements which is right adjoint to the functor taking S-ideals. This adjunction restricts to an equivalence of the category of completely regular S-frames and a full subcategory of completely regular frames. As an application of the latter equivalence, we construct the Stone-? ech compactification of a completely regular S-frame, that is, its compact coreflection in the category of completely regular S-frames.

A distinguishing feature of the study of partial frames is that a small collection of axioms of an elementary nature allows one to do much that is traditional at the level of frames or locales and of uniform or nearness frames. The axioms are sufficiently general to include as examples of partial frames bounded distributive lattices, σ-frames, κ-frames and frames.     

Direct Constructions of the Paracompact Coreflections of Frames

We present a general construction from which one can derive in a unified manner (i) the Stone–ech compactification of a frame, (ii) the Lindelöf coreflection for both frames and nearness frames and (iii) the paracompact coreflection in the category of frames.     

Ring ideals and the Stone-?ech compactification in pointfree topology

This paper shows that the compact completely regular coreflection in the category of frames is given by the frame of Jacobson radical ideals of the ring RL of real-valued continuous functions on L, as an alternative to its familiar representations in terms of (i) the l-ideals of RL as lattice-ordered ring or (ii) the ideals of the bounded part of RL which are closed in the usual uniform topology. Further, in analogy with this, the compact zero-dimensional coreflection will also be described in terms of ring ideals, this time of the ring ZL of integer-valued continuous functions on L.     

Epicompletion in Frames with Skeletal Maps, II: Compact Normal Joinfit Frames

An epireflection ψ is constructed of the category $\mathfrak{KNArS}$ of compact normal joinfit frames, with skeletal maps, in the subcategory $\mathfrak{SPArS}$ consisting of strongly projectable $\mathfrak{KNArS}$ -objects. The construction is achieved via a pushout in the category $\mathfrak{FrmS}$ of frames with skeletal maps, and involves rather intimately the regular coreflection of the object to be reflected. Further, if the regular coreflection ρ is applied to the reflection map ψ _A :A?→?ψA one obtains the extension of ρA to its absolute.     

Complete nearness σ-frames

We show that complete strong nearness σ-frames are exactly the cozero parts of complete separable strong Lindelöf nearness frames. We also relate nearness σ-frames and metric σ-frames and show that every metric σ-frame admits an admissible nearness such that it is complete as a metric σ-frame if and only if it is complete in this admissible nearness.     

Thoughts on Quotient-fine Nearness Frames

We characterize nearness frames whose completions are fine (we call them quotient-fine), and show that the subcategory QfNFrm they form is reflective in the category of strong nearness frames. The resulting functor commutes with the completion functor. QfNFrm is isomorphic to the subcategory of the functor category (RegFrm) ² given by the dense onto \(h\colon M\to L\), where 2 denotes the category with only two objects and exactly one morphism between them.     

On preparacompactness of uniform frames

We present a pointfree characterization of paracompactness via strong Cauchy completeness. We also provide a filter characterization of separability in uniform frames and determine those uniform frames that have a Lindelöf and compact completion using the notion of preparacompactness. Further, as an application of preparacompactness, we provide filter conditions for the Lindelöfness of the Hewitt realcompactification υL of a completely regular frame L.     

Enough Regular Cauchy Filters for Asymmetric Uniform and Nearness Structures

Quasi-nearness biframes provide an asymmetric setting for the study of nearness; in Frith and Schauerte (Quaest Math 33:507–530, 2010) a completion (called a quasi-completion) was constructed for such structures and in Frith and Schauerte (Quaest Math, 2012) completeness was characterized in terms of the convergence of regular Cauchy bifilters. In this paper questions of functoriality for this quasi-completion are considered and one sees that having enough regular Cauchy bifilters plays an important rôle. The quasi-complete strong quasi-nearness biframes with enough regular Cauchy bifilters are seen to form a coreflective subcategory of the strong quasi-nearness biframes with enough regular Cauchy bifilters. Here a significant difference between the symmetric and asymmetric cases emerges: a strong (even quasi-uniform) quasi-nearness biframe need not have enough regular Cauchy bifilters. The Cauchy filter quotient leads to further characterizations of those quasi-nearness biframes having enough regular Cauchy bifilters. The fact that the Cauchy filter quotient of a totally bounded quasi-nearness biframe is compact shows that any totally bounded quasi-nearness biframe with enough regular Cauchy bifilters is in fact quasi-uniform. The paper concludes with various examples and counterexamples illustrating the similarities and differences between the symmetric and asymmetric cases.     

Coreflective hull of finite strong L-topological spaces

This paper is concerned with the interaction between the logic features of the table of truth values and categorical properties of L-topological spaces and L-co-topological spaces. On one hand, it is shown that for each unital quantale L, the category of Alexandroff strong L-co-topological spaces is the coreflective hull of finite strong L-co-topological spaces. On the other hand, in the case that the quantale L is the unit interval [0,1] equipped with a continuous t-norm, it is shown that the category of Alexandroff strong [0,1]-topological spaces is the coreflective hull of finite strong [0,1]-topological spaces if and only if the continuous t-norm is an ordinal sum of the ?ukasiewicz t-norm whose set of idempotent elements is a well-ordered subset of [0,1] under the usual order.     

A bitopological point-free approach to compactifications

We study structures called d-frames which were developed by the last two authors for a bitopological treatment of Stone duality. These structures consist of a pair of frames thought of as the opens of two topologies, together with two relations which serve as abstractions of disjointness and covering of the space. With these relations, the topological separation axioms regularity and normality have natural analogues in d-frames. We develop a bitopological point-free notion of complete regularity and characterise all compactifications of completely regular d-frames. Given that normality of topological spaces does not behave well with respect to products and subspaces, probably the most surprising result is this: The category of d-frames has a normal coreflection, and the Stone-?ech compactification factors through it. Moreover, any compactification can be obtained by first producing a regular normal d-frame and then applying the Stone-?ech compactification to it. Our bitopological compactification subsumes all classical compactifications of frames as well as Smyth?s stable compactification.     

Cartesian-closed coreflective subcategories of uniform spaces generated by classes of metric spaces

The main result, in Theorem 3, is that in the category Unif of Hausdorff uniform spaces and uniformly continuous maps, the coreflective hulls of the following classes are cartesian-closed: all metric spaces having no infinite uniform partition, all connected metric spaces, all bounded metric spaces, and all injective metric spaces.Furthermore, Theorems 1 and 4 imply that if $\text{C}$ is any coreflective, cartesian-closed subcategory of Unif in which enough function space structures are finer than the uniformity of uniform convergence (as in the above examples), then either (1) $\text{C}$ is a subclass of the locally fine spaces, or (2) $\text{C}$ contains all injective metric spaces and $\text{C}$ is a subclass of the coreflective hull of all uniform spaces having no infinite uniform partition.