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.     

A Method for Constructing Coreflections for Nearness Frames

We provide a fairly general method, which is straightforward and widely applicable, for constructing some coreflections in the category of nearness frames. The method captures all coreflective subcategories with 1???1 coreflection maps; this includes the well-known uniform, totally bounded and separable coreflections. The primary application of this method answers in the affirmative the question of Dube and Mugochi [15] as to whether strong nearness frames are coreflective in nearness frames. We show that the strong coreflection can change the underlying frame, in contrast to Dube and Mugochi’s almost uniform coreflection in the category of interpolating nearness frames. The method also finds application in categories other than nearness frames, for instance, prenearness frames and nearness σ-frames. We conclude with an application to the unstructured setting where we recover the regular and completely regular coreflections in frames.     

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.     

CAUCHY POINTS OF UNIFORM AND NEARNESS FRAMES

Abstract

The notion of Cauchy point (= regular Cauchy filter) and the corresponding Cauchy spectrum, for a nearness frame (= uniform without the star-refinement condition) are investigated in various directions, including basic motivation, several functorial aspects, and the recognition of the Cauchy spectrum as the ordinary spectrum of the completion, after the unique existence of the latter is obtained as a central new result in this context.     

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.     

THE WALLMAN COMPACTIFICATION OF FRAMES

Abstract

We show that a B-conjunctive frame L, where B is a normal base for L gives rise to a strong inclusion on L and therefore a compactification of L. The resulting compact regular frame corresponds to the quotient frame obtained by Johnstone in his construction of the Wallman compactification for frames. It is also shown that, in the presence of pseudocompactness the Wallman compactification and the Wallman realcompactification coincide.     

A NOTE ON COMPLETE REGULARITY AND NORMALITY

Abstract

In this paper we introduce the notions of uniform complete regularity and uniform normality for nearness frames with the view of obtaining new and pointfree proofs of some known topological results.     

Epimorphisms of uniform frames

It is shown that some familiar properties of epimorphisms in the category of frames carry over to the categories of uniform and complete uniform frames. This is achieved by suitably enriching certain frame homomorphisms to uniform frame homomorphisms.     

STRONG ZERO-DIMENSIONALITY OF BIFRAMES AND BISPACES

Abstract

A bispace is called strongly zero-dimensional if its bispace Stone—?ech compactification is zero—dimensional. To motivate the study of such bispaces we show that among those functorial quasi—uniformities which are admissible on all completely regular bispaces, some are and others are not transitive on the strongly zero-dimensional bispaces. This is in contrast with our result that every functorial admissible uniformity on the completely regular spaces is transitive precisely on the strongly zero-dimensional spaces.

We then extend the notion of strong zero-dimensionality to frames and biframes, and introduce a De Morgan property for biframes. The Stone—Cech compactification of a De Morgan biframe is again De Morgan. In consequence, the congruence biframe of any frame and the Skula biframe of any topological space are De Morgan and hence strongly zero-dimensional. Examples show that the latter two classes of biframes differ essentially.     

Cauchy Complete Nearness Spaces

A nearness space is Cauchy complete if every regular Cauchy filter on the space is convergent. We show that the category CCNear ₂ of Cauchy complete N ₂ spaces is reflective in the category Near ₂ C of N ₂-spaces and Cauchy maps and that the reflection of an N ₂-space is given by the strict extension associated with regular Cauchy filters on the space.     

Concerning some variants of C-embedding in pointfree topology

We study three types of quotient maps of frames which are closely related to C- and C^?-quotient maps. We call them C₁-, strong C₁-, and uplifting quotient maps. C₁-quotient maps are precisely those whose induced ring homomorphisms contract maximal ideals to maximal ideals. We show that every homomorphism onto a frame is a C₁-, a strong C₁-, or an uplifting quotient map iff the frame is pseudocompact, compact, or almost compact and normal, respectively. These quotient maps are used to characterize normality and also certain weaker forms of normality in a manner akin to the characterization of normal frames as those for which every closed quotient map is a C-quotient map. Under certain conditions, we show that the Stone extension of a quotient map is C₁-, strongly C₁- or uplifting if the map has the corresponding property.     

Strong Cauchy completeness in uniform frames

We show that for uniform frames, with the underlying frame being Boolean, uniform paracompactness and strong Cauchy completeness are equivalent conditions. Certain aspects of uniform paracompactness are also considered. We then introduce the pointfree notion of preparacompactness and show that the completion of a preparacompact uniform frame is strongly Cauchy complete. We also formulate pointfree filter characterizations of Lindelöfness for regular frames in analogy to their classical topological counterparts.     

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.     

On some parallelism between complete regularity and zero-dimensionality

We explore some parallelism between the categories CRFrm and 0DFrm of completely regular frames and zero-dimensional frames, respectively, with a view to establishing zero-dimensional analogues of C*-quotients. A lattice homomorphism between the cozero parts of two completely regular frames can be lifted to a frame homomorphism between the Stone-?ech compactifications of the frames involved [13]. Here we lift a lattice homomorphism ψ: BL → BM between the Boolean parts of two zero-dimensional frames to a frame homomorphism between their universal zero-dimensional compactifications, and then study some properties of the lift.     

Some aspects of (non) functoriality of natural discrete covers of locales

Abstract

The frame S_c(L) generated by closed sublocales of a locale L is known to be a natural Boolean (“discrete”) extension of a subfit L; also it is known to be its maximal essential extension. In this paper we first show that it is an essential extension of any L and that the maximal essential extensions of L and S_c(L) are isomorphic. The construction S_c is not functorial; this leads to the question of individual liftings of homomorphisms L → M to homomorphisms S_c(L) → S_c(M). This is trivial for Boolean L and easy for a wide class of spatial L, M . Then, we show that one can lift all h : L → 2 for weakly Hausdor? L (and hence the spectra of L and S_c(L) are naturally isomorphic), and finally present liftings of h : L → M for regular L and arbitrary Boolean M.     

A note on weakly Lindelöf frames

The notion of σ^?-properness of a subset of a frame is introduced. Using this notion, we give necessary and su?cient conditions for a frame to be weakly Lindelöf. We show that a frame is weakly Lindelöf if and only if its semiregularization is weakly Lindelöf. For a completely regular frame L, we introduce a condition equivalent to weak realcompactness based on maximal ideals of the cozero part of L. This enables us to show that every weakly realcompact almost P -frame is realcompact. A new characterization of weakly Lindelöf frames in terms of neighbourhood strongly divisible ideals of ?? is provided. The closed ideals of ?? equipped with the uniform topology are applied to describe weakly Lindelöf frames.     

Pointfree forms of Dowker’s and Michael’s insertion theorems

In this paper we prove two strict insertion theorems for frame homomorphisms. When applied to the frame of all open subsets of a topological space they are equivalent to the insertion statements of the classical theorems of Dowker and Michael regarding, respectively, normal countably paracompact spaces and perfectly normal spaces. In addition, a study of perfect normality for frames is made.     

SUPERCATEGORIES OF THE CATEGORY OF NEARNESS SPACES

Abstract

The framework in which nearness spaces were defined by H. Herrlich [1] and [2], leads one to consider the supercategory Pow of the category Near of nearness spaces, having as objects all pairs (X,ξ), where X is a set and ξ ? P(P(X)) is any subset of the power set of the power set of X, and as morphisms f: (X,ξ) → (Y,n) all functions f: X → Y such that, if A ? ξ then fA □ {f(A) | A ξ A} ? η. In this paper we show that the full subcategories of Pow comprising the objects satisfying subsets of the prenearness space axioms lie in a lattice of bireflections or bicoreflections. This serves as a first step towards the aim of characterizing all bireflective (resp. bicoreflective) and even all initially complete subcategories of Pow.     

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.     

Remarks on the frame envelope of a σ-frame

The familiar equivalence between σ-frames and σ-coherent frames, given by the frame envelopes of σ-frames, is shown to induce an equivalence between stably continuous σ-frames and stably continuous frames. Similarly, the analogue of the former for σ-biframes is proved to provide an equivalence between compact regular σ-biframes and compact regular biframes. As an application we obtain the equivalence between stably continuous σ-frames and compact regular σ-biframes due to Matutu as an easy consequence of its frame counterpart established earlier by Banaschewski and Brümmer. This provides an affirmative answer to a question posed by Dana Scott.