Hereditary Coreflective Subcategories of Categories of Topological Spaces

Let be an epireflective subcategory of the category Top of topological spaces that is not bireflective (e.g., the category of Hausdorff spaces, the category of Tychonoff spaces) and ℬ be a coreflective subcategory of . Extending the corresponding result obtained for coreflective subcategories of Top we prove that ℬ is hereditary if and only if it is closed under the formation of prime factors. As a consequence we obtain that every hereditary coreflective subcategory ℬ of containing a non-discrete space is generated by a class of prime spaces and if is a quotient-reflective subcategory of Top, then the assignment gives a bijection of the collection of all hereditary coreflective subcategories of Top that contain the class FG of all finitely generated spaces onto the collection of all hereditary coreflective subcategories of that contain . Some applications of these results in the categories of Hausdorff spaces, Tychonoff spaces and zero-dimensional Hausdorff spaces are presented.Mathematics Subject Classifications (2000) 18D15, 54B30.     

On Hereditary Coreflective Subcategories of Top

Let A be a topological space which is not finitely generated and CH(A) denote the coreflective hull of A in Top. We construct a generator of the coreflective subcategory SCH(A) consisting of all subspaces of spaces from CH(A) which is a prime space and has the same cardinality as A. We also show that if A and B are coreflective subcategories of Top such that the hereditary coreflective kernel of each of them is the subcategory FG of all finitely generated spaces, then the hereditary coreflective kernel of their join CH(AB) is again FG.     

Heredity and Coreflective Subcategories of the Category of Topological Spaces

Every hereditary coreflective subcategory of Top containing the category of finitely-generated spaces is shown to be generated by a class of spaces having a unique accumulation point. It is also shown that the coreflective hull of a union of two hereditary coreflective subcategories of Top need not be hereditary so that a coreflective subcategory of Top need not have a hereditary coreflective kernel.     

Productivity of Coreflective Subcategories of Semitopological Groups

The present paper generalizes to semitopological and quasitopological groups some results achieved by Horst Herrlich and the second author for topological groups. The results concern preserving products in coreflective subcategories. Unlike in paratopological or topological groups, there are non-finitely productive bicoreflective subcategories of quasitopological groups. We desctribe bicoreflective subcategories of semitopological groups that are either finitely productive or productive or their productivity number is submeasurable. To achieve the results, several general factorization theorems for maps on products will be proved with the help of modification of NSS property of groups.     

Gorenstein子范畴与相对奇点范畴

令A是阿贝尔范畴, T是A的一个自正交子范畴, 且T中每个对象均有有限投射维数和内射维数. 假设左Gorenstein子范畴lG(T)等于T的右正交类,且右Gorenstein子范畴rG(T)等于T的左正交类,我们证明了Gorenstein子范畴$G(T)$等于T的左正交类与T的右正交类之交,并且证明了它们的稳定范畴三角等价于A关于T的相对奇点范畴.作为应用,令$R$是有有限左自内射维数的左诺特环, $_RC_s$是半对偶化双模,且所有内射左$R$-模的平坦维数的上确界有限, 我们证明了 若$\mbox{}_RC$有有限内射（平坦）维数且$C$的右正交类包含$R$,则存在从$C$-Gorenstein投射模与关于$C$的Bass类的交到关于$C$-投射模的相对奇点范畴间的三角等价,推广了某些经典的结果.     

Subcategories of Filter Tower Spaces

The concept of a convergence tower space, or equivalently, a convergence approach space is formulated here in the context of a Cauchy setting in order to include a completion theory. Subcategories of filter tower spaces are defined in terms of axioms involving a general t-norm, T, in order to include a broad range of spaces. A T-regular sequence for a filter tower space is defined and, moreover, it is shown that the category of T-regular objects is a bireflective subcategory of all filter tower spaces. A completion theory for subcategories of filter tower spaces is given.     

Classifying Substructures of Extriangulated Categories via Serre Subcategories

Applied Categorical Structures - We give a classification of substructures (= closed subbifunctors) of a given skeletally small extriangulated category by using the category of defects, in a...     

图表示范畴的两个子范畴

引进图表示范畴的两个子范畴,研究它们的同调性质。     

因素空间和范畴

本文讨论了几种模糊范畴的Ｔｏｐｏｓ性质，把集丛理论、范畴理论应用到因素空间的研究之中，建立了同一因素集上和不同因素集上的因素空间之间的关系，并给出了子因素空间的概念和子因素空间的布尔运算。     

对称的满层L-Kent收敛空间范畴的子范畴

在满层的L-Kent收敛空间中引入了对称性的概念,定义了对称的满层L-Kent收敛空间范畴,对称的满层L-极限空间范畴,对称的满层L-主收敛空间范畴,对称的满层L-拓扑空间范畴.证明这四个范畴是拓扑范畴,并且后一个是前一个的反射子范畴.最后证明了对称的满层L-Kent收敛空间范畴和对称的满层L-极限空间范畴是笛卡儿闭的.     

Static Subcategories of the Module Category of a Finite-Dimensional Hereditary Algebra

Let k be a field, Λ a finite-dimensional hereditary k-algebra, and modΛ the category of all finite-dimensional Λ-modules. We are going to characterize the representation type of Λ (tame or wild) in terms of the possible subcategories statM of all M-static modules, where M is an indecomposable Λ-module.     

Classifying Dense Resolving and Coresolving Subcategories of Exact Categories Via Grothendieck Groups

Classification problems of subcategories have been deeply considered so far. In this paper, we discuss classifying dense resolving and dense coresolving subcategories of exact categories via their Grothendieck groups. This study is motivated by the classification of dense triangulated subcategories of triangulated categories due to Thomason.     

Hereditary,Additive and Divisible Classes in Epireflective Subcategories of Top

Hereditary coreflective subcategories of an epireflective subcategory A of Top such that I ₂ ? A (here I ₂ is the two-point indiscrete space) were studied in [4]. It was shown that a coreflective subcategory B of A is hereditary (closed under the formation subspaces) if and only if it is closed under the formation of prime factors. The main problem studied in this paper is the question whether this claim remains true if we study the (more general) subcategories of A which are closed under topological sums and quotients in A instead of the coreflective subcategories of A. We show that this is true if A ? Haus or under some reasonable conditions on B. E.g., this holds if B contains either a prime space, or a space which is not locally connected, or a totally disconnected space or a non-discrete Hausdorff space. We touch also other questions related to such subclasses of A. We introduce a method extending the results from the case of non-bireflective subcategories (which was studied in [4]) to arbitrary epireflective subcategories of Top. We also prove some new facts about the lattice of coreflective subcategories of Top and ZD.     

On the Extension of Seminormal Functors to the Category of Tychonoff Spaces

Chigogidze proposed a construction of extending a normal functor from the category Comp to the category Tych. We can apply his scheme to seminormal functors and study the properties of the original functor which are preserved under extension. We introduce the concept of functor having an invariant extension from Comp to Tych because the very existence of this invariance plays a key role in the preservation of the properties of a seminormal functor in its extension. It is proved that the superextension functor λ has an invariant extension. We check that if a seminormal functor has an invariant extension then its extension preserves a point, the empty set, intersection and is a monomorphic functor. If this functor has finite degree then its extension is continuous and hence a seminormal functor in Tych. If the functor is of infinite degree then continuity may be lost. Namely, we show that the extension of λ for Tych is not continuous.     

正规弱θ空间的无限Tychonoff积

本文证明:(1)如果X=∏_σ∈∑X_σ是|∑|-仿紧空间,则X是正规弱θ可加空间当且仅当?F∈[∑]^<ω,∏_σ∈FX_σ是正规弱θ-可加空间.(2)设X=∏_i∈ωX_i是可效仿紧的,则下列三条等价:是正规弱θ-可加的;?F∈[ω]^<ω,∏_i∈FX_i是正规弱θ-可加的;?n∈ω;∏_i≤n     

Compactness in Categories of S-Net Spaces

The $S$ -net spaces studied are convergence structures whose convergences are expressed by using generalized nets, the so called $S$ -nets, which are obtained from the usual nets by replacing the category of directed sets and cofinal maps with an arbitrary construct $S$ . We investigate compactness in categories of $S$ -net spaces defined by introducing continuous maps in a natural way and imposing some usual convergence axioms.     

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.     

Tensor Products in Categories of Topological Spaces

In this paper symmetric monoidal closed structures on coreflective subcategories of the category of (Hausdorff) topological spaces are studied. We describe all such structures on the category of (Hausdorff) pseudoradial spaces and some of its subcategories and give an example of a coreflective subcategory of the category of Hausdorff topological spaces admitting a proper class of symmetric monoidal closed structures.     

Injective and Epi-Projective Objects in Categories of Convex Spaces

We give a characterization of injective and epi-projective objects in some categories of convex spaces.