Compactness, Perfectness, Separation, Minimality and Closedness with Respect to Closure Operators

In this paper, the characterization of closed and strongly closed subobjects of an object in categories of various types of filter convergence spaces is given and it is shown that they induce a notion of closure. Furthermore, each of the notions of compactness, perfectness, separation, minimality and absolute closedness with respect to these two new closure operators are characterized in these categories and some known results are re-obtained.     

拉回正合范畴中小子对象的探讨

拉回正合范畴是Abelian范畴的真正推广,是界在正合范畴与Abelian范畴之间的一类范畴.本文利用拉回-推出,引进拉回正合范畴的小子对象概念,并给出小子对象相关的性质以及等价条件.     

On classes of relations and graphs determined by subobjects and factorobjects

The classes of relations and graphs determined by subobjects and factorobjects are studied. We investigate whether such classes are closed under products, whether they are finitely generated by products and subobjects and whether a class can be described alternatively by subobjects and factorobjects. This is related to good characterizations.     

A perturbation result for q-open quotient morphisms in normed spaces and applications to linear relations

Using some techniques of perturbation theory for quotient morphisms between Banach spaces we obtain necessary and sufficient conditions for the stability of the topological index of a q-open quotient morphism under small (with respect to the gap topology) perturbations with quotient morphisms. As an application we obtain necessary and sufficient conditions for the stability of the topological index of an open linear relation with closed multivalued part between normed spaces under small perturbations with linear relations.     

Computable Real-Valued Functions on Recursive Open and Closed Subsets of Euclidean Space

In this paper we study intrinsic notions of “computability” for open and closed subsets of Euclidean space. Here we combine together the two concepts, computability on abstract metric spaces and computability for continuous functions, and delineate the basic properties of computable open and closed sets. The paper concludes with a comprehensive examination of the Effective Riemann Mapping Theorem and related questions.     

A Note on the Distributivity of the Huq Commutator Over Finite Joins

We show that the Huq commutator distributes over finite joins, in any semi-abelian algebraically cartesian closed category. As a consequence we show that for semi-abelian varieties of universal algebras (more generally for semi-abelian categories with large directed colimits of subobjects preserved, for each object B, by the functor B×??), the distributivity of the Huq commutator over joins is equivalent to algebraic cartesian closedness.     

New torsion theory in unital abelian <Emphasis Type="Italic">ℓ</Emphasis>-groups

This paper introduces the notion of a functorial torsion class (FTC): in a concrete category \mathfrakC\mathfrak{C} which has image factorization, one considers monocoreflective subcategories which are closed under formation of subobjects.     

An `Unsitely" Result on Atomic Morphisms

We give an `elementary" proof, without mentioning sites, that any section of an atomic geometric morphism is open, and any section of a connected atomic morphism is an open surjection. Previously, these results were known only for bounded morphisms. As a by-product, we obtain a proof that any connected atomic morphism with a section is necessarily bounded.     

Definable categories

We introduce the notion of a definable category–a category equivalent to a full subcategory of a locally finitely presentable category that is closed under products, directed colimits and pure subobjects. Definable subcategories are precisely the finite-injectivity classes. We prove a 2-duality between the 2-category of small exact categories and the 2-category of definable categories, and provide a new proof of its additive version. We further introduce a third vertex of the 2-category of regular toposes and show that the diagram of 2-(anti-)equivalences between three 2-categories commutes; the corresponding additive triangle is well-known.     

关于转移开、闭映射的注记

介绍转移闭值[转移开值]映射及其相关的概念,说明它们之间的相互关系,然后得出若干个转移闭值[转移开值]的性质.     

Around the Proof of the Legendre–Cauchy Lemma on Convex Polygons

We briefly describe the history of the proofs of the well-known Cauchy lemma on comparison of the distances between the endpoints of two convex open polygons on a plane or sphere, present a rather analytical proof, and explain why the traditional constructions lead in general to inevitable appearance of nonstrictly convex open polygons. We also consider bendings one to the other of two isometric open or closed convex isometric polygons.     

拓扑空间子集的最小开集和最大闭集

讨论对于拓扑空间子集A是否存在包含A的最小开集和是否存在包含于A的最大闭集问题,证明了拓扑空间X是一个T1空间之充分且必要条件是,对于X的每一个子集A,X中存在包含A的最小开集（存在包含于A的最大闭集）当且仅当A是X的开集（闭集）,同时给出几个例子说明了定理的条件.     

Bjarni Jónsson’s contributions in algebra: an addendum

Normal categories are pointed categorical counterparts of 0-regular varieties, i.e., varieties where each congruence is uniquely determined by the equivalence class of a fixed constant 0. In this paper, we give a new axiomatic approach to normal categories, which uses self-dual axioms on a functor defined using subobjects of objects in the category. We also show that a similar approach can be developed for 0-regular varieties, if we replace subobjects with subsets of algebras containing 0.     

Closed mappings onto metric spaces

We investigate the classes of spaces that can be mapped onto a metrizable space by a closed mapping with fibers having a given property $\text{P}$. We give some conditions which assure that such classes are closed under the action of perfect or open and compact mappings. Such a treatment includes the investigation of paracompact p-spaces and M-spaces. We also discuss spaces that can be mapped onto a metacompact Moore space.     

Coverings of 3-manifolds by open balls and two open solid tori

Hempel and McMillan showed that a closed 3-manifold that can be covered by three open balls is a connected sum of S³- and S²-bundles over S¹. In this paper we obtain a classification of all closed 3-manifolds that can be covered by two open balls and one open solid torus or by one open ball and two open solid tori.     

Embedding locales and formal topologies into positive topologies

A positive topology is a set equipped with two particular relations between elements and subsets of that set: a convergent cover relation and a positivity relation. A set equipped with a convergent cover relation is a predicative counterpart of a locale, where the given set plays the role of a set of generators, typically a base, and the cover encodes the relations between generators. A positivity relation enriches the structure of a locale; among other things, it is a tool to study some particular subobjects, namely the overt weakly closed sublocales. We relate the category of locales to that of positive topologies and we show that the former is a reflective subcategory of the latter. We then generalize such a result to the (opposite of the) category of suplattices, which we present by means of (not necessarily convergent) cover relations. Finally, we show that the category of positive topologies also generalizes that of formal topologies, that is, overt locales.     

关于次中紧空间

本文得到如下主要结果：一个空间是次中紧的当且仅当它的每令定向开覆盖有σ-闭包保持闭加细使得它被一切紧子集组成的族所加细；次中紧空间在闭的紧覆盖映射下的象是次中紧的；一个空间是可度量的当且仅当它是次中紧的Moore空间。     

低碳经济背景下企业技术创新模式决策机制研究——基于碳税政策视角

为探究低碳经济背景下企业技术创新模式的决策机制,基于低碳技术创新的开放式创新和封闭式创新两种模式,运用演化经济学理论和方法构建企业低碳技术创新模式选择博弈模型,在策略选择的演化稳定性分析基础上,通过数值模拟演示不同决策参数和初始条件下模型的演化结果。结果表明:企业选择开放式低碳技术创新模式的前提条件是当开放式低碳技术创新策略的竞争性收益大于投入成本且系统中选择该种模式的初始企业比例达到一定水平;企业技术吸收能力、开放式低碳技术创新成本和碳税征收强度对低碳经济背景下企业技术创新模式的选择均有重要的影响;为促进企业实施开放式低碳技术创新,政府决策应将碳税征收强度保持在合理范围内,同时重视信息化平台搭建助力企业间知识交流和共享,营造企业实施开放式低碳技术创新的良好环境。     

On Pure Quotients and Pure Subobjects

In the theory of accessible categories, pure subobjects, i.e. filtered colimits of split monomorphisms, play an important role. Here we investigate pure quotients, i.e., filtered colimits of split epimorphisms. For example, in abelian, finitely accessible categories, these are precisely the cokernels of pure subobjects, and pure subobjects are precisely the kernels of pure quotients.     

Ideals and Clots in Pointed Regular Categories

We clarify the relationship between ideals, clots, and normal subobjects in a pointed regular category with finite coproducts. Partially supported by South African NRF Research partially supported by the Hungarian National Foundation for Scientific Research grant no. K61007.