拓扑分子格的分离公理

在[1]中我们建立了拓扑分子格的理论,它既是古典的点集拓扑学的推广,又是晚近发展起来的Fuzzy拓扑学的推广,对于某些Fuzzy格L(如L是线性序集或L是分子格等),它也是L—Fuzzy拓扑学的推广。因此,凡在拓扑分子格中得到的结果自然都是上述各种拓扑学中相应定理的一般化形式。在本文中我们将讨论拓扑分子格的分离公理。 我们熟知点集拓扑学中的分离公理有多种不同的等价形式。以正则性为例,设X是拓扑空间,X叫正则的,当且仅当对每个点a∈X以及a的每个开邻域U,a有开邻域V满足条件V~-U。这一分离公理又可表述为:设a∈X,F是X中不包含a的闭集,则有开集P     

A note on Cauchy spaces

In previous papers, various notions of (strongly) closed subobject, (strongly) open subobject, connected, compact and T _i , i=0,1,2 objects in a topological category were introduced and compared. The main objective of this paper is to characterize each of these classes of objects in the category of Cauchy spaces as well as to examine how these generalizations are related.     

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.     

On the product of homogeneous spaces

Within the class of Tychonoff spaces, and within the class of topological groups, most of the natural questions concerning ‘productive closure’ of the subclasses of countably compact and pseudocompact spaces are answered by the following three well-known results: (1) [ZFC] There is a countably compact Tychonoff space X such that X × X is not pseudocompact; (2) [ZFC] The product of any set of pseudocompact topological groups is pseudocompact; and (3) [ZFC+ MA] There are countably compact topological groups G₀, G₁ such that G₀ × G₁ is not countably compact.In this paper we consider the question of ‘productive closure” in the intermediate class of homogeneous spaces. Our principal result, whose proof leans heavily on a simple, elegant result of V.V. Uspenski?, is this: In ZFC there are pseudocompact, homogeneous spaces X₀, X₁ such that X₀ × X₁ is not pseudocompact; if in addition MA is assumed, the spaces X_i may be chosen countably compact.Our construction yields an unexpected corollary in a different direction: Every compact space embeds as a retract in a countably compact, homogeneous space. Thus for every cardinal number α there is a countably compact, homogeneous space whose Souslin number exceeds α.     

Maximal element theorems in product FC-spaces and generalized games

Let I be a finite or infinite index set, X be a topological space and (Yi,{φNi})_i∈I be a family of finitely continuous topological spaces (in short, FC-space). For each i∈I, let be a set-valued mapping. Some existence theorems of maximal elements for the family {Ai}_i∈I are established under noncompact setting of FC-spaces. As applications, some equilibrium existence theorems for generalized games with fuzzy constraint correspondences are proved in noncompact FC-spaces. These theorems improve, unify and generalize many important results in recent literature.     

On fuzzy covering properties

The purpose of this paper is to introduce properties of the notion of α-compactness for fuzzy topological spaces. Moreover, α _c-compact spaces are introduced and properties of them are also discussed for fuzzy topological spaces.      

C-scattered fuzzy topological spaces

In this paper, we define the concept of C-scattered fuzzy topological spaces and obtain some properties about them. In particular, we study the relation between C-scattered spaces and its fuzzy extension, it is proved that C-scattered fuzzy topological spaces are invariant by fuzzy perfect maps, and that, in the realm of paracompact fuzzy topological spaces, the C-scattered spaces verify that their product by other fuzzy spaces is also paracompact fuzzy.     

On separation axioms in fuzzy topological spaces,fuzzy neighborhood spaces,and fuzzy uniform spaces

A certain number of separation axioms for fuzzy topological spaces are provided, all of which are good extensions of the topological (T₀), (T₁), or (T₂). All valid implications between the different axioms are studied and counterexamples are given for the nonvalid ones.     

Spectrum topology of a residuated lattice

In this paper, given a non-commutative residuated lattice L, a topological space is constructed using certain fuzzy subsets of L. Indeed, we show that the set of all prime fuzzy filters of a non-commutative residuated lattice L forms a topological space. Particularly, we show that this space is compact and a T ₀-space and its certain subspaces are Hausdorff spaces. Finally, we show that the set of all prime filters of L is also a Hausdorff space.     

An extremal problem for finite topologies and distributive lattices

Let (r₁, r₂, …) be a sequence of non-negative integers summing to n. We determine under what conditions there exists a finite distributive lattice L of rank n with r_i join-irreducibles of rank i, for all i = 1, 2, …. When L exists, we give explicit expressions for the greatest number of elements L can have of any given rank, and for the greatest total number of elements L can have. The problem is also formulated in terms of finite topological spaces.     

On countably {\mu}-paracompact spaces

A topological space X is countably paracompact if and only if X satisfies the condition (A): For any decreasing sequence {F_i} of non-empty closed sets with \({\bigcap_{i=1}^{\infty} F_{i} = \emptyset}\) there exists a sequence {G_i} of open sets such that \({\bigcap_{i=1}^{\infty}\overline{G_{i}}=\emptyset}\) and \({F_{i} \subset G_{i}}\) for every i. We will show, by an example, that this is not true in generalized topological spaces. In fact there is a \({\mu}\)-normal generalized topological space satisfying the analogue of A which is not even countably \({\mu}\)-metacompact. Then we study the relationships between countably \({\mu}\)-paracompactness, countably \({\mu}\)-metacompactness and the condition corresponding to condition A in generalized topological spaces.     

Almost compactness in fuzzy topological spaces

We introduce and study almost compactness for fuzzy topological spaces. We show that the almost continuous image of an almost compact fuzzy topological space is almost compact. Moreover, we show that generally almost compactness for fuzzy topological spaces is not product-invariant, but if X and Y are almost fuzzy topological spaces and X is product related to Y, then their fuzzy topological product is almost compact.     

Applications of the extent analysis method on fuzzy AHP

In this paper, a new approach for handling fuzzy AHP is introduced, with the use of triangular fuzzy numbers for pairwise comprison scale of fuzzy AHP, and the use of the extent analysis method for the synthetic extent value S_i of the pairwise comparison. By applying the principle of the comparison of fuzzy numbers, that is, V(M₁ ⩾ M₂) = 1 iff m₁ ⩾ m₂, V(M₂ ⩾ M₁) = hgt(M₁ ∩ M₂) = μ_M1 (d), the vectors of weight with respect to each element under a certaine criterion are represented by d(A_i) = min V(S_i ⩾ S_k), k = 1, 2,…, n; k ≠ i. This decision process is demonstrated by an example.     

Fuzzy closed and fuzzy perfect mappings

It is shown that a fuzzy continuous map f is fuzzy perfect iff f × i_z is fuzzy closed for every fuzzy topological space Z.     

Some comments on fuzzy variables

Nahmias introduced the concept of a fuzzy variable as a possible axiomatic framework from which a rigorous theory of fuzziness may be constructed. In this paper we attempt to shed more light on fuzzy variables in analogy with random variables. In particular, we study the problem: if X₁, X₂,…,X_n are mutually unrelated fuzzy variables with common membership function μ and α₁, α₂,…,α_n are real numbers satisfying α_i ? o for every i and Σ_i=1ⁿα_i=1, when does does Z = Σ_{i = 1}ⁿα_iX_i have the same membership function μ?     

m-Uniformization and metrization of fuzzy topological spaces

In this paper the fuzzy pseudo metric is introduced in the fuzzy unit interval and based on it theorems on m-uniformization and metrization of fuzzy topological spaces are proved.     

Uniform approximation by some Hermite interpolating splines

In this paper some upper bound for the error ∥ s-f ∥_∞ is given, where f ε C¹[a,b], but s is a so-called Hermite spline interpolant (HSI) of degree 2q ?1 such that f(x_i) = s(x_i), f′(rmx_i) = s′(x_i), s^(j) (x_i) = 0 (i = 0, 1, …, n; j = 2, 3, …, q ?1; n > 0, q > 0) and the knots x_i are such that a = x₀ < x₁ < … < x_n = b. Necessary and sufficient conditions for the existence of convex HSI are given and upper error bound for approximation of the function fε C¹[a, b] by convex HSI is also given.     

On Q-sobriety

The study of fixed-basis variety-based topology was initiated by S.A. Solovyov (in 2008), which, among other things, generalizes fuzzy topology. We extend within this framework, an earlier result due to Srivastava et al. (in 1998), which showed that the category of sober fuzzy topological spaces is the epireflective hull of the fuzzy Sierpinski space in the category of T₀-fuzzy topological spaces.