Abstract:
In this note, we obtain some improvements of results established onδ-semicontinuous functions in [3] and show that a function f:(X,τ) → (Y, σ) is almost δ-semicontinuous if and only if f:(X, τ_s) → (Y, σ_s) is semi-continuous, where τ_s and σ_s are the semiregularizations of τ and σ, respectively.

Abstract:
The aim of this paper is to introduce the concepts of somewhat slightly generalized double fuzzy semicontinuous functions and somewhat slightly generalized double fuzzy semiopen functions in double fuzzy topological spaces. Some interesting properties and characterizations of these functions are introduced and discussed. Furthermore, the relationships among the new concepts are discussed with some necessary examples. 1. Introduction In 1968, Chang [1] was the first to introduce the concept of fuzzy topological spaces. These spaces and their generalization are later developed by Goguen [2], who replaced the closed interval by more general lattice . On the other hand, by the independent and parallel generalization of Kubiak and ？ostak’s [3, 4], made topology itself fuzzy besides their dependence on fuzzy set in 1985. Various generalizations of the concept of fuzzy set have been done by many authors. In [5–10], Atanassove introduced the notion of intuitionistic fuzzy sets. Later ？oker [11] defined intuitionistic fuzzy topology in Chang’s sense. Then, Mondal and Samanta [12] introduced the intuitionistic gradation of openness of fuzzy sets. Gutiérrez García and Rodabaugh [13], in 2005, replaced the term “intuitionistic” and concluded that the most appropriate work is under the name “double.” In 1980, Jain [14] introduced the notion of slightly continuous functions. On the other hand, Nour [15] defined slightly semicontinuous functions as a weak form of slight continuity and investigated their properties. In [16], Noiri introduced the concept of slightly -continuous functions. Sudha et al. [17] introduced slightly fuzzy -continuous functions. Also in 2004, Ekici and Caldas [18] introduced the notion of slight -continuity (slight -continuity). In this paper, the concepts of somewhat slightly generalized double fuzzy semicontinuous functions and somewhat slightly generalized double fuzzy semiopen functions are introduced. Several interesting properties and characterizations are introduced and discussed. Furthermore, the relationships among the concepts are obtained and established with some interesting counter examples. 2. Preliminaries Throughout this paper, let be a nonempty set, the unit interval , , and . The family of all fuzzy sets in is denoted by . is the family of all fuzzy points in . By and we denote the smallest and the greatest fuzzy sets on . For a fuzzy set , denotes its complement. Given a function , and defined the direct image and the inverse image of , defined by and for each , , and , respectively. All other notations are standard notations

Abstract:
We consider general integral functionals on the Sobolev spaces of multiple valued functions, introduced by Almgren. We characterize the semicontinuous ones and recover earlier results of Mattila as a particular case. Moreover, we answer positively to one of the questions raised by Mattila in the same paper.

Abstract:
In this paper, we apply the notion of pre-$mathcal{I}$-open sets to present and study a new class of functions called contra pre-$mathcal{I}$-continuous functions in ideal topological spaces. Relationships between this new class and other classes of functions are investigated and some characterisations of this new class of functions are studied. Also we introduce the notions of contra strong $beta$-$mathcal{I}$-continuous functions and contra $delta$-$mathcal{I}$-continuous functions to obtain decompositions of contra $alpha$-$mathcal{I}$-continuity and contra semi-$mathcal{I}$-continuity.

Abstract:
We present a decomposition of two topologies which characterize the upper and lower semicontinuity of the limit function to visualize their hidden and opposite roles with respect to the upper and lower semicontinuity and consequently the continuity of the limit. We show that (from the statistical point of view) there is an asymmetric role of the upper and lower decomposition of the pointwise convergence with respect to the upper and lower decomposition of the sticking convergence and the semicontinuity of the limit. This role is completely hidden if we use the whole pointwise convergence. Moreover, thanks to this mirror effect played by these decompositions, the statistical pointwise convergence of a sequence of continuous functions to a continuous function in one of the two symmetric topologies, which are the decomposition of the sticking topology, automatically ensures the convergence in the whole sticking topology. 1. Introduction Since the end of the nineteenth century several outstanding papers appeared to formulate a set of conditions, which are both necessary and sufficient, to be added to pointwise convergence of a sequence of continuous functions, to preserve continuity of the limit. Indeed, all classical kinds of convergences of sequences of functions between metric spaces (Dini, Arzelà, Alexandroff) are based on the pointwise convergence assumption that has been always considered a preliminary one. Recently, in [1, 2], Caserta et al. proposed a new model to investigate convergences in function spaces: the statistical one. Actually, they obtained results parallel to the classical ones, concerning the continuity of the limit, in spite of the fact that statistical convergence has a minor control of the whole set of functions. In [2] they proved that continuity of the limit of a sequence of functions is equivalent to several modes of statistical convergence which are similar, but weaker than the classical ones. A parallel to the classical results is expected since, after all, in [3] the authors found the statistical convergence to be the same as a very special regular triangular matrix summability method for bounded (and some unbounded) sequences. Thus many new results concerning statistical convergence follow from the corresponding known results for matrix summability. In 1969 Bouleau [4, 5] defined the sticking topology as the weakest topology finer than pointwise convergence to preserve continuity. In [6], Beer presented two new topologies on , finer than the topology of pointwise convergence, which are indeed the decomposition of the sticking

Abstract:
New characterizations of almost contra-precontinuity are presented. These characterizations are used to develop a new weak form of almost contra-precontinuity. This new weak form is then used to extend several results in the literature concerning almost contra-precontinuity.

Abstract:
It is shown in this paper that two positive elements of a C*-algebra agree on all lower semicontinuous traces if and only if they are equivalent in the sense of Cuntz and Pedersen. A similar result is also obtained in the more general case where the two elements are comparable by their values on the lower semicontinuous traces. This result is used to give a characterization of the functions on the cone of lower semicontinuous traces of a stable C*-algebra that arise from positive elements of the algebra.

Abstract:
We give an extension to a nonconvex setting of the classical radial representation result for lower semicontinuous envelope of a convex function on the boundary of its effective domain. We introduce the concept of radial uniform upper semicontinuity which plays the role of convexity, and allows to prove a radial representation result for nonconvex functions. An application to the relaxation of multiple integrals with constraints on the gradient is given.

Abstract:
In 1989 Ganster and Reilly [6] introduced and studied the notion of LC-continuous functions via the concept of locally closed sets. In this paper we consider a stronger form of LC-continuity called contra-continuity. We call a function f:(X, ) ￠ ’(Y, ) contra-continuous if the preimage of every open set is closed. A space (X, ) is called strongly S-closed if it has a finite dense subset or equivalently if every cover of (X, ) by closed sets has a finite subcover. We prove that contra-continuous images of strongly S-closed spaces are compact as well as that contra-continuous, 2-continuous images of S-closed spaces are also compact. We show that every strongly S-closed space satisfies FCC and hence is nearly compact.