A note on sequentially compact quasi-pseudo-metric spaces

It is asked in [3] whether sequential compactness is equivalent to compactness in a quasi-pseudo-metric space. In this note we give a counterexample and a proposition relating to this question.     

On hyperspace topologies via distance functionals in quasi-metric spaces

Summary Let (X,d) be a quasi-pseudo-metric space. We investigate hyperspace topologies on P₀(X) defined by distance functionals. In particular, the K-topology is introduced and compared with other hyperspace topologies. Some properties of the Wijsman topology and the K-topology are explored.     

Wijsman and Hit-and-Miss Topologies of Quasi-Metric Spaces

The relationship between the Wijsman topology and (proximal) hit-and-miss topologies is studied in the realm of quasi-metric spaces. We establish the equivalence between these hypertopologies in terms of Urysohn families of sets. Our results generalize well-known theorems and provide easier proofs. In particular, we prove that for a quasi-pseudo-metrizable space (X,T) the Vietoris topology on the set P ₀(X) of all nonempty subsets of X is the supremum of all Wijsman topologies associated with quasi-pseudo-metrics compatible with T. We also show that for a quasi-pseudo-metric space (X,d) the Hausdorff extended quasi-pseudo-metric is compatible with the Wijsman topology on P ₀(X) if and only if d ^–1 is hereditarily precompact.     

Many Familiar Categories can be Interpreted as Categories of Generalized Metric Spaces

The simple concepts of (general) distance function and homometry (a map that preserves distances up to a calibration) are introduced, and it is shown how some natural distance functions on various mathematical objects lead to concrete embeddings of the following categories into the resulting category DIST°: quasi-pseudo-metric, topological, and (quasi-)uniform spaces with various kinds of maps; groups and lattice-ordered abelian groups; rings and modules, particularly fields; sets with reflexive relations and relation-preserving maps (particularly directed loop-less graphs and quasi-ordered sets); measured spaces with Radon-continuous maps; Boolean, Brouwerian, and orthomodular lattices; categories with combined objects, for example topological groups, ordered topological spaces, ordered fields, Banach spaces with linear contractions or linear continuous maps and so on.     

Completeness in generalized ultrametric spaces

We analyze the relationship between four notions of completeness for Γ-ultrametric spaces. The notions we consider include Cauchy completeness, strong Cauchy completeness, spherical completeness and injectivity. In the process we show that the category of Γ-ultrametric spaces is equivalent to the category of flabby separated presheaves on Γ ^op .     

Banach压缩映象原理与空间完备性

本文进一步揭示了Ｂａｎａｃｈ压缩映象原理与完备性的关系，指出：一般地，Ｂａｎａｃｈ压缩映象原理等价于道路完备性，在一定条件下才等价于完备性，进一步给出了这一等价性成立的充要条件     

On the completeness of a metric related to the Bergman metric

We study the completeness of a metric which is related to the Bergman metric of a bounded domain (sometimes called the Burbea metric or Fuks metric). We provide a criterion for its completeness in the spirit of the Kobayashi criterion for the completeness of the Bergman metric. In particular we prove that in hyperconvex domains our metric is complete.     

On Strict Extensions of Nearness Spaces

Strict extensions of nearness spaces are constructed as spaces of round Cauchy filters. Morita's simple extension is identified as the strict extension generated by Morita-generated filters. Carlson's B-completeness is compared with Herrlich completeness and completeness of Morita T-uniformities. The three completeness concepts are shown to be equivalent in regular nearness spaces.     

Some Characterizations of Completeness for Trellises in Terms of Joins of Cycles

This paper gives some new characterizations of completeness for trellises by introducing the notion of a cycle-complete trellis. One of our results yields, in particular, a characterization of completeness for trellises of finite length due to K. Gladstien (see K. Gladstien: Characterization of completeness for trellises of finite length, Algebra Universalis 3(1973), 341–344).     

Weakly Cauchy Filters and Quasi-Uniform Completeness

It is well-known that the notion of a Smyth complete quasi-uniform space provides an appropriate notion of completeness to study many interesting quasi-metric spaces which appear in theoretical computer science. We observe that several of these spaces actually possess a stronger form of completeness based on the use of weakly Cauchy filters in the sense of H. H. Corson and we develop a theory of completion and completeness for this kind of filters. In parallel, we also study a more general notion of completeness based on the use of certain stable filters. Thus our results extend and generalize important theorems of Á. Császár, J. R. Isbell and N. R. Howes on uniform completeness.     

On completeness of locally-internal categories

Locally-internal categories over a topos E are regarded as categories enriched in the bicategory Span E. In this paper we develop some aspects of completeness. For locally-internal categories, completeness means: finite completeness over each fiber, transition functors along the maps of the base topōs, and a Beck-Chevalley condition (see Bénabou, C.R. Acad. Sci. Paris 281 (1975) A897–900). We prove that this notion can be obtained by particularizing to Span E the general notion of completeness of enriched category theory, given in terms of indexed limits. We give also an adjoint functor theorem.     

Completeness of security markets and solvability of linear backward stochastic differential equations

For a standard Black-Scholes type security market, completeness is equivalent to the solvability of a linear backward stochastic differential equation (BSDE, for short). An ideal case is that the interest rate is bounded, there exists a bounded risk premium process, and the volatility matrix has certain surjectivity. In this case the corresponding BSDE has bounded coefficients and it is solvable leading to the completeness of the market. However, in general, the risk premium process and/or the interest rate could be unbounded. Then the corresponding BSDE will have unbounded coefficients. For this case, do we still have completeness of the market? The purpose of this paper is to discuss the solvability of BSDEs with possibly unbounded coefficients, which will result in the completeness of the corresponding market.     

Problem of checking the functional completeness in the class of piecewise parallel functions

The author showed previously that it is possible to implement an arbitrary piecewise parallel function by circuits of linear elements and one nonlinear element implementing the θ-function. Thus, the functional completeness of a particular system of functions in the class of piecewise parallel functions was established. In this paper we consider the same class and study its completeness property for a system of functions whose closures contain all linear functions. We formulate and prove a criterion of functional completeness for such systems in terms of precomplete classes.     

Clifton-pohl torus and geodesic completeness by a ‘complex’ point of view

We show that a natural complexification and a mild generalization of the idea of completeness guarantee geodesic completeness of Clifton-Pohl torus; we explicitely compute all of its geodesics.     

全函数空间和置换空间的一些性质

In this paper, an equivalent relation among the reflexivity, weak sequential completeness and bounded completeness in full function space is given. Some results on weakly sequential compactness of subset and the property （u） in substitution spaces are obtained.     

On the completeness of the system of eigenvectors of electromagnetic waveguides

Several statements of the spectral problem in the theory of regular waveguides are considered. The completeness of the system of eigenvectors of these problems, which is a consequence of the earlier established completeness property of a certain system of eigenvectors, is proved.     

Banach压缩映象原理与空间完备性

本文揭示了Banach压缩映象原理与空间完备性的关系,证明了在具有一致局部连通性的距离空间中,压缩映象原理与空间完备性仍然等价.     

Proof-functional connectives and realizability

The meaning of a formula built out of proof-functional connectives depends in an essential way upon the intensional aspect of the proofs of the component subformulas. We study three such connectives, strong equivalence (where the two directions of the equivalence are established by mutually inverse maps), strong conjunction (where the two components of the conjunction are established by the same proof) and relevant implication (where the implication is established by an identity map). For each of these connectives we give a type assignment system, a realizability semantics, and a completeness theorem. This form of completeness implies the semantic completeness of the type assignment system.     

有关一致空间完备性证明的修正

设μ为集X的最大一致结构,对于一致空间（X,μ）的完备性,指出有关文献中的证明存在疏漏,并直接运用一致空间完备性的定义给出相关结论的证明．