Compact supported endographs and fuzzy sets

A metric is defined on a space of functions from a locally compact metric space X into the unit interval I in terms of the Hausdorff metric distance between their compact supported endographs in X × I. Convergence in this metric is shown to be equivalent to the conjunction of the Hausdorff metric convergence of supports in X and two conditions involving numerical values of the functions. The space of nonempty compact subsets of X with the Hausdorff metric is imbedded in the above function space by the characteristic function on subsets of X. Applications of these results to fuzzy subsets of X and fuzzy dynamical systems on X are indicated.     

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.     

Descriptions of equivalence of O-connectivity

We first study some properties of the subspace, and investigate into the relationship of separation between a fuzzy topological space (fts) and its subspace. Then we obtain the equivalence conditions for O-connectivity. The results on O-connectivity and separation are very similar to those in general topology. Finally we discuss the relationship of connectivity between an O-connected set A in the fts (X, ω ($\text{T}$)) induced by the crisp topological space (X, $\text{T}$) and the crisp set A₀ (=supp A) in (X, $\text{T}$).     

On fuzzy almost continuous convergence in fuzzy function spaces

In this paper, we study the fuzzy almost continuous convergence of fuzzy nets on the set FAC(X, Y) of all fuzzy almost continuous functions of a fuzzy topological space X into another Y. Also, we introduce the notions of fuzzy splitting and fuzzy jointly continuous topologies on the set FAC(X, Y) and study some of its basic properties.     

Weakly mixing property and chaos

In this paper, we define and study strong Kato chaos for a group action on a compact metric space. Let X be a compact metric space without isolated points, and let G be a topologically commutative group on X. If the dynamical system (X, G) is weakly mixing, then it is chaotic in the strong sense of Kato.     

A class of linear differential dynamical systems with fuzzy matrices

This paper investigates the first order linear fuzzy differential dynamical systems with fuzzy matrices. We use a complex number representation of the α-level sets of the fuzzy system, and obtain the solution by employing such representation. It is applicable to practical computations and has also some implications for the theory of fuzzy differential equations. We then present some properties of the 2-dimensional dynamical systems and their phase portraits. Some examples are considered to show the richness of the theory and we can clearly see that new behaviors appear. We finally present some conclusions and new directions for further research in the area of fuzzy dynamical systems.     

Integrals,conditional expectations,and martingales of multivalued functions

Let (Ω, $\text{A}$, μ) be a finite measure space and $\text{X}$ a real separable Banach space. Measurability and integrability are defined for multivalued functions on Ω with values in the family of nonempty closed subsets of $\text{X}$. To present a theory of integrals, conditional expectations, and martingales of multivalued functions, several types of spaces of integrably bounded multivalued functions are formulated as complete metric spaces including the space L¹(Ω; $\text{X}$) isometrically. For multivalued functions in these spaces, multivalued conditional expectations are introduced, and the properties possessed by the usual conditional expectation are obtained for the multivalued conditional expectation with some modifications. Multivalued martingales are also defined, and their convergence theorems are established in several ways.     

Solution of fuzzy equations with extended operations

Assuming that 1 is any operation defined on a product set X × Y and taking values on a set Z, it can be extended to fuzzy sets by means of Zadeh's extension principle. Given a fuzzy subset C of Z, it is here shown how to solve the equation $\text{A 1 B = C}$ (or $\text{A 1 B ? C}$) when a fuzzy subset A of X (or a fuzzy subset B of Y) is given. The methodology we provide includes, as a special case, the resolution of fuzzy arithmetical operations, i.e. when 1 stands for +, ?, × or ÷, extended to fuzzy numbers (fuzzy subsets of the real line). The paper is illustrated with several examples in fuzzy arithmetic.     

Singular homology groups of fuzzy topological spaces

Let L be a completely distributive lattice with order reversing involution, and (X, τ) an L-fuzzy topological space. The purpose of this paper is to introduce the fundamental concept of fuzzy algebraic topology-the singular homology groups of the L-fuzzy topological space, in such a way that they take the (usual) cubical singular homology groups of a topological space as a special case. Also, we shall prove that they are L-fuzzy homeomorphic invariants.     

Inducing sensitivity on hyperspaces

Let (X,d) be a compact metric space and (K(X),dH) be the space of all non-empty compact subsets of X equipped with the Hausdorff metric dH. The dynamical system (X,f) induces another dynamical system . We study the relations between the various forms of sensitivity of the systems (X,f) and . We prove that all forms of sensitivity of partly imply the same for (X,f), and the converse holds in some cases.     

Representation and approximation by series of continuous functions

Let X be a topological space and let C(X) be the ring of all real-valued continuous functions defined on X. We study the representation and approximation of continuous functions by sums of infinite series. Among other results, we give sufficient conditions in order to represent or approximate every continuous function by infinite series of functions, belonging to a previously fixed subfamily of C(X), when X is either a locally compact paracompact space or a Lindelöf space.     

On the existence of continuous (approximate) roots of algebraic equations

The present paper considers the existence of continuous roots of algebraic equations with coefficients being continuous functions defined on compact Hausdorff spaces. For a compact Hausdorff space X, C(X) denotes the Banach algebra of all continuous complex-valued functions on X with the sup norm ∥⋅_∥∞. The algebra C(X) is said to be algebraically closed if each monic algebraic equation with C(X) coefficients has a root in C(X). First we study a topological characterization of a first-countable compact (connected) Hausdorff space X such that C(X) is algebraically closed. The result has been obtained by Countryman Jr, Hatori-Miura and Miura-Niijima and we provide a simple proof for metrizable spaces.Also we consider continuous approximate roots of the equation zn−f=0 with respect to z, where f∈C(X), and provide a topological characterization of compact Hausdorff space X with dimX?1 such that the above equation has an approximate root in C(X) for each f∈C(X), in terms of the first ?ech cohomology of X.     

Covering properties on X2⧹Δ, W-sets,and compact subsets of Σ-products

Let Δ ? X¹ be the diagonal. In the first part of this paper, we show that a compact space X is Corson compact (resp., Eberlein compact; compact metric) if and only if X²?Δ is metalindelöf (resp., σ-metacompact; paracompact). In the second part of the paper, we investigate the notion of a W-set in a space X, which is defined in terms of an infinite game. We show that a compact space X is Corson compact if and only if X has a W-set diagonal, and that a compact scattered space X is strong Eberlein compact if and only if each point of X is a W-set in X.     

On completion of fuzzy metric spaces

Completions of fuzzy metric spaces (in the sense of George and Veeramani) are discussed. A complete fuzzy metric space Y is said to be a˜fuzzy metric completion of a˜given fuzzy metric space X if X is isometric to a˜dense subspace of Y. We present an example of a˜fuzzy metric space that does not admit any fuzzy metric completion. However, we prove that every standard fuzzy metric space has an (up to isometry) unique fuzzy metric completion. We also show that for each fuzzy metric space there is an (up to uniform isomorphism) unique complete fuzzy metric space that contains a˜dense subspace uniformly isomorphic to it.     

Compactifications whose remainders are zero dimensional and retract

LetX be a locally compact non compact space. Necessary and sufficient conditions forfX/X to be a retract offX are given wherefX is the Freudenthal compactification ofX. LetX be a locally compact and zero dimensional space,m be any cardinal number andJ be a set with cardinalitym. It is proved thatX has a dyadic family of powerm if and only if there exist and compactificationY ofX such thatY/X=2 ^J andY/X is a retract ofY.     

Michael's problem and weakly infinite-dimensional spaces

Let X be a compact Hausdorff space. Suppose that any multivalued map , where Y is a Gδ subset of a Banach space, such that the values of F are convex and closed in Y, has a continuous single-valued selection. Then we prove that X is weakly infinite-dimensional. This provides a partial solution of Gδ-problem, posed by Ernest Michael.     

CH, a problem of Rolewicz and bidiscrete systems

We give a construction under CH of a non-metrizable compact Hausdorff space K such that any uncountable ‘nice’ semi-biorthogonal sequence in C(K) must be of a very specific kind. The space K has many nice properties, such as being hereditarily separable, hereditarily Lindelöf and a 2-to-1 continuous preimage of a metric space, and all Radon measures on K are separable. However K is not a Rosenthal compactum.We introduce the notion of a bidiscrete system in a compact space K. These are subsets of K² which determine biorthogonal systems of a special kind in C(K) that we call nice. We note that for every infinite compact Hausdorff space K, the space C(K) has a bidiscrete system and hence a nice biorthogonal system of size d(K), the density of K.     

A new bound on the cardinality of homogeneous compacta

We show (in ZFC) that if X is a compact homogeneous Hausdorff space then |X|?²t(X), where t(X) denotes the tightness of X. It follows that under GCH the character and the tightness of such a space coincide.     

Lipschitz compact operators

We introduce the notion of Lipschitz compact (weakly compact, finite-rank, approximable) operators from a pointed metric space X into a Banach space E. We prove that every strongly Lipschitz p-nuclear operator is Lipschitz compact and every strongly Lipschitz p-integral operator is Lipschitz weakly compact. A theory of Lipschitz compact (weakly compact, finite-rank) operators which closely parallels the theory for linear operators is developed. In terms of the Lipschitz transpose map of a Lipschitz operator, we state Lipschitz versions of Schauder type theorems on the (weak) compactness of the adjoint of a (weakly) compact linear operator.     

Fixed points for fuzzy mappings

In this paper the problem of the existence and computation of fixed points for fuzzy mappings is approached. A fuzzy mapping R over a set X is defined to be a function attaching to each x in X a fuzzy subset R_χ of X. An element x of X is called fixed point of R iff its membership degree to R_χ is at least equal to the membership degree to R_χ of any y?X, i.e. R_χ(χ)? R_χ(y)(?y?X). Two existence theorems for fixed points of a fuzzy mapping are proved and an algorithm for computing approximations of such a fixed point is described. The convergence theorem of our algorithm is proved under the restrictive assumption that for any x in X, the membership function of R_χ has a ‘complementary function’. Examples of fuzzy mappings having this property are given, but the problem of proving general criteria for a function to have a complementary remain open.