The aim of this paper is to investigate the general approximation structure, weak approximation operators, and Pawlak algebra in the framework of fuzzy lattice, lattice topology, and auxiliary ordering. First, we prove that the weak approximation operator space forms a complete distributive lattice. Then we study the properties of transitive closure of approximation operators and apply them to ...

In this note we prove that the weak topology of a Banach space has the Continuous Representability Property which means that every (weakly) continuous total preorder defined on a Banach space can be represented by a weakly continuous utility function. This shows that weak topologies are perfect to look for continuous utility functions that represent preferences on infinite-dimensional commodity...

For a strictly stationary sequence of regularly varying random variables we study functional weak convergence of partial sum processes in the space D[0, 1] with the Skorohod J1 topology. Under the strong mixing condition, we identify necessary and sufficient conditions for such convergence in terms of the corresponding extremal index. We also give conditions under which the regular variation pr...

For n ≥ 3, let Ω ⊂ R be a bounded smooth domain and N ⊂ R be a compact smooth Riemannian submanifold without boundary. Suppose that {un} ⊂ W (Ω, N) are weak solutions to the perturbed n-harmonic map equation (1.2), satisfying (1.3), and uk → u weakly in W (Ω, N). Then u is an n-harmonic map. In particular, the space of n-harmonic maps is sequentially compact for the weak-W 1,n topology. §

A topological group G is called extremely amenable if every continuous action of G on a compact space has a fixed point. This concept is linked with geometry of high dimensions (concentration of measure). We show that a von Neumann algebra is approximately finite-dimensional if and only if its unitary group with the strong topology is the product of an extremely amenable group with a compact gr...

let g = (v,e) be a locally finite graph, i.e. a graph in which every vertex has finitely many adjacent vertices. in this paper, we associate a topology to g, called graphic topology of g and we show that it is an alexandroff topology, i.e. a topology in which intersec- tion of every family of open sets is open. then we investigate some properties of this topology. our motivation is to give an e...

This paper introduces a new approach to topology, based on category theory and universal algebra, and called categorically-algebraic (catalg) topology. It incorporates the most important settings of lattice-valued topology, including poslat topology of S.~E.~Rodabaugh, $(L,M)$-fuzzy topology of T.~Kubiak and A.~v{S}ostak, and $M$-fuzzy topology on $L$-fuzzy sets of C.~Guido. Moreover, its respe...

In this paper, we have proved results similar to Tychonoff's Theorem on embedding a space of functions with the topology pointwise convergence into Tychonoff product topological spaces, but applied function C(X,Y) all continuous from X uniform Y family subsets and (weak) set-open topology. We also investigated following question: how is related algebraic structures (such as groups, rings vector...

The term CW-complex comes from “closure-finite with the weak topology”, where “closure-finite” refers to A3 and “weak topology” refers to A4. A finite complex is a CW-complex with only finitely many cells. Observe that if X is a finite complex, A4 is redundant, since W is the union of the compact sets φα((φ n α) −1W ), and these are closed since X is Hausdorff. If X has cells of dimension n but...