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.     

A note on precompact uniform frames

Precompactness or total boundedness for uniform frames is usually distinguished by a cover approach. In this note, we provide alternate characterizations of precompact uniform frames. In particular, we formulate pointfree filter analogues of various classical topological results on precompactness. We also revisit the notion of convergence and clustering of filters in a frame and introduce weakly Cauchy filters and strong Cauchy completeness in the setting of uniform 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.     

A Shirota Theorem for Frames

We establish a version of the Shirota Theorem which characterises realcompactness in terms of completeness for frames; an interesting problem considering the various notions of realcompactness for frames and the fact that the frame completion behaves differently from its spatial counterpart. Amongst other consequences of this characterisation, we can describe the completion of a uniform sigma frame. We highlight some differences between frames and spaces when considering the Lindelöf property, realcompactness, paracompactness and completeness. Most of these results appear in the doctoral thesis of the author.     

On Cauchy Homomorphisms of Nearness Frames

To include the pointfree case, the notion of a Cauchy map easily extends as the condition that the preimage of any regular Cauchy filter contains a regular Cauchy filter. This extension, however, can be unsatisfactory if the regular Cauchy filters are scarce or non-existent, making the condition too weak, indeed sometimes even void. In this paper a variant of the notion, independent on the existence of any kind of filters, is studied: a homomorphism is called fully Cauchy if it lifts to the completions. This is generally stronger than, and in the spatial and metric cases it coincides with the previously mentioned property. Moreover, it is stable under localization.     

E-Compactness in Pointfree Topology

We present a pointfree analogue of E-compactness as introduced by Engelking and Mrówka. In particular, we show that a frame L is a closed quotient of a copower of a regular frame E iff L is complete in its E-nearness. We further show that the version of E-compactness that is more compatible with the classical one corresponds to Cauchy completeness in its E-nearness.     

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.     

Frames with Transitive Structures

A classical result in the theory of uniform spaces is that any topological space with a base of clopen sets admits a uniformity with a transitive base and the uniform topology of such a space has a base of clopen sets. This paper presents a pointfree generalization of this, both to uniform and quasi-uniform frames, together with various properties concerning total boundedness, compactifications and completions.     

数学分析中若干具有共性理论的问题

探讨数学分析中若干具有共性理论的问题,并重点阐述数学分析中若干关于一致问题与Cauchy定理的教学探索和实践.     

Lower and upper regularizations of frame semicontinuous real functions

As discovered recently, Li and Wang’s 1997 treatment of semicontinuity for frames does not faithfully reflect the classical concept. In this paper we continue our study of semicontinuity in the pointfree setting. We define the pointfree concepts of lower and upper regularizations of frame semicontinuous real functions. We present characterizations of extremally disconnected frames in terms of these regularizations that allow us to reprove, in particular, the insertion and extension type characterizations of extremally disconnected frames due to Y.-M. Li and Z.-H. Li [Algebra Universalis 44 (2000), 271–281] in the right semicontinuity context. It turns out that the proof of the insertion theorem becomes very easy after having established a number of basic results regarding the regularizations. Notably, our extension theorem is a much strengthened version of Li and Li's result and it is proved without making use of the insertion theorem. The first and second named authors acknowledge financial support from the Ministry of Education and Science of Spain and FEDER under grant MTM2006-14925-C02-02. The first named author also acknowledges financial support from the University of the Basque Country under grant UPV05/101. The third named author acknowledges financial support from the Centre of Mathematics of the University of Coimbra/FCT.     

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.     

Approximation Spaces in the Numerical Analysis of Operator Equations

We show that generalized approximation spaces can be used to prove stability and convergence of projection methods for certain types of operator equations in which unbounded operators occur. Besides the convergence, we also get orders of convergence by this approach, even in case of non-uniformly bounded projections. We give an example in which weighted uniform convergence of the collocation method for an easy Cauchy singular integral equation is studied.     

Choiceless, Pointless, but not Useless: Dualities for Preframes

We provide the appropriate common ‘(pre)framework’ for various central results of domain theory and topology, like the Lawson duality of continuous domains, the Hofmann–Lawson duality between continuous frames and locally compact sober spaces, the Hofmann–Mislove theorems about continuous semilattices of compact saturated sets, or the theory of stably continuous frames and their topological manifestations. Suitable objects for the pointfree approach are quasiframes, i.e., up-complete meet-semilattices with top, and preframes, i.e., meet-continuous quasiframes. We introduce the pointfree notion of locally compact well-filtered preframes, show that they are just the continuous preframes (using a slightly modified definition of continuity) and establish several natural dualities for the involved categories. Moreover, we obtain various characterizations of preframes having duality. Our results hold in ZF set theory without any choice principles.      

On Strong Inclusions and Asymmetric Proximities in Frames

The strong inclusion, a specific type of subrelation of the order of a lattice with pseudocomplements, has been used in the concrete case of the lattice of open sets in topology for an expedient definition of proximity, and allowed for a natural pointfree extension of this concept. A modification of a strong inclusion for biframes then provided a pointfree model also for the non-symmetric variant. In this paper we show that a strong inclusion can be non-symmetrically modified to work directly on frames, without prior assumption of a biframe structure. The category of quasi-proximal frames thus obtained is shown to be concretely isomorphic with the biframe based one, and shown to be related to that of quasi-uniform frames in a full analogy with the symmetric case.     

Stanley's zrank conjecture on skew partitions

We present an affirmative answer to Stanley's zrank conjecture, namely, the zrank and the rank are equal for any skew partition. We show that certain classes of restricted Cauchy matrices are nonsingular and furthermore, the signs are determined by the number of zero entries. We also give a characterization of the rank in terms of the Giambelli-type matrices of the corresponding skew Schur functions. Our approach also applies to the factorial Cauchy matrices and the inverse binomial coefficient matrices.

     

一致结构的单子及其应用

讨论了乘积一致结构的单子的刻画 .利用一致结构的单子给出了 Cauchy网及含小集集族的非标准特征 .作为应用给出一致空间几个重要定理的离散化证明 .     

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.     

Functorial Quasi-Uniformities on Frames

We present a unified study of functorial quasi-uniformities on frames by means of Weil entourages and frame congruences. In particular, we use the pointfree version of the Fletcher construction, introduced by the authors in a previous paper, to describe all functorial transitive quasi-uniformities.Mathematics Subject Classifications (2000) 06D22, 54B30, 54E05, 54E15, 54E55.Jorge Picado: The authors acknowledge partial financial assistance by the Centro de Matemática da Universidade de Coimbra/FCT.     

On Compactness of Frames

We define the notions of extension-closed and nearly closed sublocales of frames and present, amongst other things, a pointfree and choice-free proof of a result of Harris characterizing extension-closed subspaces. We also characterize compact frames in terms of the latter form of closedness.     

Local Connectedness Made Uniform

The uniformly locally connected reflection for a locally connected uniform frame is constructed. Applications of this construction to the theory of locally connected completely regular frames are given. One such application is that if a completely regular frame is locally connected and pseudocompact then every compactification of it is locally connected.