Coreflective hull of finite strong L-topological spaces

This paper is concerned with the interaction between the logic features of the table of truth values and categorical properties of L-topological spaces and L-co-topological spaces. On one hand, it is shown that for each unital quantale L, the category of Alexandroff strong L-co-topological spaces is the coreflective hull of finite strong L-co-topological spaces. On the other hand, in the case that the quantale L is the unit interval [0,1] equipped with a continuous t-norm, it is shown that the category of Alexandroff strong [0,1]-topological spaces is the coreflective hull of finite strong [0,1]-topological spaces if and only if the continuous t-norm is an ordinal sum of the ?ukasiewicz t-norm whose set of idempotent elements is a well-ordered subset of [0,1] under the usual order.     

On adjunctions between Lim, SL-Top, and SL-Lim

Consider (L,*,1) be a commutative, strictly two-sided quantale with the underlying lattice L being meet-continuous. Two adjunctions, one is between limit spaces and stratified L-limit spaces and the other is between stratified L-limit spaces and stratified L-topological spaces, are established. The first adjunction can be viewed as an extension of Lowen's adjunction between the category of topological spaces and stratified [0,1]-topological spaces. The second is an extension of an adjunction between limit spaces and (stratified) L-topological spaces established in U. Höhle and T. Kubiak (Höhle-Kubiak, Semigroup Forum (2007)).     

Nonstandard extensions of uniform algebraic systems

Important applications of nonstandard analysis to the theory of Banach spaces are based on the construction of the nonstandard hull of a normed linear space [1,2]. By making use of the iterated nonstandard enlargements [3], in the present article we propose a universal construction for arbitrary uniform algebraic systems which allows one to study the nonstandard hulls and completions of such systems.In Section 1 we give necessary information about nonstandard enlargements and prove a number of important results in the general theory of monads. In Section 2 we present basic facts on nonstandard topologies and uniform algebraic systems. In Section 3 we describe a general algebraic construction with the help of which we study the question of completing uniform algebraic systems. Conditions for existence of nonstandard hulls for such systems are obtained in Section 4.Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 35, No. 5, pp. 1094–1105, September–October, 1994.     

Observations on Irreducible Sets and Sober Spaces

This paper, using the monads theory in the topological space, gives a new characterization of irreducible sets in the nonstandard enlarged models. Further, the discretization expression of Sober topological spaces is presented.     

On -topological vector spaces generated by a co-tower of L-topological vector spaces

In this paper, a characterization for an I(L)-topological space to be generated by a given co-tower of L-topological spaces is obtained. Moreover, the relationship between some properties of an I(L)-topological vector space generated by a co-tower of L-topological vector spaces and the corresponding properties of the given co-tower of L-topological vector spaces is investigated. Our results show that if an I(L)-topological vector space generated by a co-tower of L-topological vector spaces has some properties, such as local convexity and local boundedness, then all L-topological vector spaces in the co-tower also have the same properties. But the converse is incorrect even in the case of I-topological vector space generated by a co-tower of classical topological vector spaces. Finally, we supply a necessary and sufficient condition for an I(L)-topological vector space generated by a co-tower of L-topological vector spaces with some properties, such as local convexity and local boundedness, to have such properties too.     

Separation properties of induced I(L)-topological spaces

The concept of induced I(L)-topological spaces has been introduced by Kubiak (Ph.D. Thesis, UAM, Poznan, 1985) and independently by Wang (Kexue Tongbao 34 (5) (1989) 333). In this paper, the separation properties in the sense of Hutton–Reilly of induced I(L)-topological spaces are investigated. The main result of the paper is a characterization of L-topological spaces by means of the appropriate Hutton–Reilly separation properties of its induced I(L)-topological space.     

On epireflective and coreflective hulls of a particular L-topological space

In this paper, we study some aspects of the category L-ZTop of zero-dimensional L-topological spaces. After noting that it is a topological category, we identify a ‘Sierpinski object’ L_Z in it. We further show that two epireflective hulls of L_Z respectively turn out to be the categories of zero-dimensional T₀-L-topological spaces and of zero-dimensional sober L-topological spaces. We also determine the coreflective hull of L_Z in the category of L-topological spaces.     

Hausdorff L-topological spaces over a base space

In this paper, we introduce the notion of Hausdorff L-topological spaces over a base space, where L is a complete lattice and give some characterizations of these spaces. And we show that the category of Hausdorff L-topological spaces over a base space is a reflective subcategory of the category of L-topological spaces over a base space.     

Completely prime L-filters,irreducible L-filters and sobriety

In this paper, for a frame L, we characterize modified sobriety in stratified L-topological spaces and strong L-topological spaces internally using certain classes of stratified L-filters. While the first characterization using completely prime L-filters is trivial and applies to all stratified L-topological spaces, the other one using irreducible L-filters generalizes an approach of R.E. Ho?mann to the lattice-valued case but is restricted to either the case that the lattice is a complete Boolean algebra or to the case of completely distributive lattices and strong L-topological spaces.     

α-拓扑空间的遗传性

在拓扑空间(X,T)中提出了α-拓扑空间(X,Tα),对(X,Tα)中的开集的判别法进行了研究,在遗传性方面得到了一些重要的结论.     

Fibrewise injectivity and Kock–Zöberlein monads

Using Escardó’s characterization of injectivity via Kock–Zöberlein monads, we introduce suitable monads in comma categories of topological spaces that yield characterizations of fibrewise injectivity in topological T0-spaces, with respect to the class of embeddings, and of dense, of flat and of completely flat embeddings. Characterizations, in the category of topological spaces, of injective maps with respect to the same classes of embeddings follow easily from the results obtained for T0-spaces. Moreover, it is shown that, together with the corresponding embeddings, injective continuous maps form a weak factorization system in the category of topological (T0-)spaces and continuous maps.     

I(L)-拓扑空间中的可数性、分离性与仿紧性

Kubiak^[9]与王戈平^[2]分别独立地引进了诱导I(L)-拓扑空间概念。本文用在[1]定义的L-拓扑空间的可数性、分离性与仿紧性等来刻画由它诱导的I(L)-拓扑空间的相应的这些性质。     

SOME NEW TYPE OF DIFFERENCE SEQUENCE SPACES DEFINED BY MODULUS FUNCTION AND STATISTICAL CONVERGENCE

In this article we introduce the difference sequence spaces W 0 [f,Δm ],W 1 [f,Δm ],W∞[f,Δm ] and S[f,Δm ],defined by a modulus function f.We obtain a relation between W 1 [f,Δm ] ∩l∞[f,Δm ] and S[f,Δm ] ∩l∞[f,Δm ] and prove some inclusion results.     

A note on the left and right translations in L-topological groups

In this note, we show by a counterexample that a proposition on L-homeomorphism property of the left and right translations in L-topological groups given by Bayoumi [On initial and final L-topological groups, Fuzzy Sets and Systems 156 (2005) 43–54] is incorrect, and offer its modified version.     

ON CERTAIN INITIAL COMPLETIONS IN FUZZY TOPOLOGY

Abstract

Following a lead given by I.W. Alderton, it is shown that the MacNeille completion and the universal initial completion coincide for the categories of zero-dimensional fuzzy T₀-topological spaces, T₀-fuzzy closure spaces, 2T ₀-fuzzy bitopological spaces, and T ₁-fuzzy topological spaces and that these turn out to be respectively the categories of zero-dimensional fuzzy topological spaces, fuzzy closure spaces, fussy bitopological spaces, and fuzzy R ₀ topological spaces.     

$ L^2 $-torsion of Hyperbolic Manifolds of Finite Volume

Suppose is a compact connected odd-dimensional manifold with boundary, whose interior M comes with a complete hyperbolic metric of finite volume. We will show that the -topological torsion of and the -analytic torsion of the Riemannian manifold M are equal. In particular, the -topological torsion of is proportional to the hyperbolic volume of M, with a constant of proportionality which depends only on the dimension and which is known to be nonzero in odd dimensions [HS]. In dimension 3 this proves the conjecture [Lü2, Conjecture 2.3] or [LLü, Conjecture 7.7] which gives a complete calculation of the -topological torsion of compact -acyclic 3-manifolds which admit a geometric JSJT-decomposition.?In an appendix we give a counterexample to an extension of the Cheeger-Müller theorem to manifolds with boundary: if the metric is not a product near the boundary, in general analytic and topological torsion are not equal, even if the Euler characteristic of the boundary vanishes. Submitted: March 1998, revised: July 1998.     

A Certain Class of Weighted Approximations for Integrable Functions and Applications

A certain class of weighted approximations, which extends the results of Masjed-Jamei [6] is introduced for integrable functions and some of upper bounds are obtained for the absolute value of the errors of such approximations in two L¹[a, b] and L^∞[a, b] spaces. As the main motivation for introducing the aforesaid class, it is shown that many new inequalities can be generated from the given error bounds. Some illustrative examples are presented in this sense. Moreover, by using the obtained error bounds, a nonstandard type of three-point weighted quadrature rules is introduced and its error bounds are computed.     

Every topological category is convenient for Gelfand duality

In this paper we generalize our work on Gelfand dualities in cartesian closed topological categories [42] to categories which are only monoidally closed. Using heavily enriched category theory we show that under very mild conditions on the base category function algebra functor and spectral space functor exist, forming a pair of adjoint functors and establishing a duality between function algebras and spectral spaces. Using recent results in connection with semitopological functors, we show that every (E,M)-topological category is endowed with at least oneconvenient monoidal structure admitting a generalized Gelfand duality. So it turns out that there is no need for a cartesian closed structure on a topological category in order to study generalized Gelfand-Naimark dualities.     

R(L)型诱导空间及连通性

本文研究了R（L）-型诱导空间，利用R（L）-值下半连续函数，讨论了R（L）型诱导空间的基本性质，得到了R（L）型诱导空间保持笛卡积和连通性，并证明了诱导映射和生成映射是可交换的。     

可加广义代数格上的 Tietze 扩张定理

可加的广义代数格范畴与 T₀ 拓扑空间范畴相等价, 从这个观点出发, 作者把可加广义代数格作为一个闭集格, 在其上建立 Urysohn 引理和 Tietze 扩张定理. 这是拓扑理论在格上的一种新推广, 有助于格上拓扑理论的研究和广义连续格理论的应用.