Abstract A collection of spanning subgraphs TS, of a graph G is said to be a spanning graph topology if it satisfies the three axioms: Nn, K0 ∈ TS where, n = |V (G)|, the collection is closed under any union and finite intersection. Let (X, T) be a topological space in point set topology and Y ⊆ X then, TY = {U ∩ Y : U ∈ T} is a topological space called a subspace topology or relative topology defined by T on Y . In this paper we discusses the subspace spanning graph topology defined by the graph topology TS on a spanning subgraph H of G.

Abstract In this article, mixed fuzzy topology and its topological properties have been studied. Mixed fuzzy topology is defined with the help of quasi-coincidence and closure of a fuzzy set in one of the fuzzy topologies. Thus, a new fuzzy topology is generated from the given two fuzzy topologies. This new fuzzy topology may or may not contain the topological properties of the parent topologies. This study identifies some topological properties that are carried to the mixed fuzzy topology from the given parent fuzzy topologies and some other properties which are not carried to the mixed fuzzy topology. Here a base for mixed fuzzy topology from the bases of the given parent topologies is constructed. Considering the regularity of one of the parent topologies mixed fuzzy topology is investigated. Hausdorff’s properties of mixed fuzzy topological spaces are also discussed. It is now of general interest to know which properties are carried to the mixed topology and which are not. A few of these are being tried to answer here in this paper.

Abstract Background teaching practice shows little development of mathematical competences in students to provide a satisfactory response to everyday situations, manifested in difficulties in identifying topology as a tool that allows modeling this type of situation. Objective to develop and implement a mathematical modeling proposal involving topology as a way of connecting the real world with mathematics. Methodology action research with a qualitative approach was carried out with fifty university students. Analysis and results: the importance of studying topological invariants is highlighted because they allow us to find differences and similarities in closed three-dimensional trajectories, elements that make up the structure of a surface. The importance of: relating metric spaces with topology; the need to handle and institutionalise a clear, precise, and proper language of topology as components that allow the student to recognise that the topological properties of knots (invariants) are directly related to the properties of the surfaces that can be generated from them.

Resumen Antecedentes la práctica docente evidencia escaso desarrollo de competencias matemáticas en los estudiantes para dar respuesta satisfactoria a situaciones cotidianas, manifiestas en dificultades para identificar la topología como una herramienta que admite modelar este tipo de situaciones. Objetivo elaborar e implementar una propuesta de modelización matemática, que involucra la topología, como manera de conectar el mundo real con las matemáticas. Metodología se adelantó una investigación-acción de enfoque cualitativo con cincuenta estudiantes universitarios. Análisis y resultados: se destaca la importancia de estudiar invariantes topológicas porque permiten encontrar diferencias y similitudes en trayectorias tridimensionales cerradas, elementos que conforman la estructura de una superficie. Se resalta la importancia de: relacionar espacios métricos con la topología; necesidad de manejar e institucionalizar un lenguaje claro, preciso y propio de topología como componentes que permiten al estudiante reconocer que las propiedades topológicas de los nudos (invariantes), están directamente relacionadas con las propiedades de las superficies que se pueden generar a partir de ellos.

Abstract In this paper, some characterizations of fuzzifying strong compactness are given, including characterizations in terms of nets and presubbases. Several characterizations of locally strong compactness in the framework of fuzzifying topology are introduced and the mapping theorems are obtained.

Abstract Several notions on soft topology are studied and their basic properties are investigated by using the concept of soft open sets and soft closure operators which are derived from the basics of soft set theory established by Molodtsov [7]. In this paper we introduce some soft separation axioms called Soft R0 and soft R1 in soft topological spaces which are defined over an initial universe with a fixed set of parameters. Many characterizations and properties of these spaces are found. Necessary and sufficient conditions for a soft topological space to be a soft Ri for i = 0, 1 space are also presented. Furthermore, the concept of functions with soft closed graph and soft cluster sets are defined. Many results on theses two concepts are proved also it is proved that a function has a soft closed graph if and only if its soft cluster set is degenerate.

Abstract Function space topologies are developed for EC(Y,Z), the class of equi-continuous mappings from a topological space Y to a uniform space Z. Properties such as splittingness, admissibility etc. are defined for such spaces. The net theoretic investigations are carried out to provide characterizations of splittingness and admissibility of function spaces on EC(Y,Z). The open-entourage topology and pointtransitive-entourage topology are shown to be admissible and splitting respectively. Dual topologies are defined. A topology on EC(Y,Z) is found to be admissible (resp. splitting) if and only if its dual is so.

Abstract Induced fuzzy infi topological space is already introduced by Saha and Bhattacharya [Saha A.K., Bhattacharya D. 2015, Normal Induced Fuzzy Topological Spaces, Italian Journal of Pure and Applied Mathematics, 34, 45-56]. In this paper for the said space, we further analyse some properties viz. fuzzy I-continuity, fuzzy infi open mappings and fuzzy infi closed mappings etc. Also we study product fuzzy infi topological space and establish some results concerned with it.

Abstract The main purpose of this article is to introduce and study some new spaces of I-convergence of triple sequences in intuitionistic fuzzy normed space defined by compact operator i.e 3SI (μ,ν)(T ) and 3SI 0(μ,ν)(T ) and examine some fundamental properties, fuzzy topology and verify inclusion relations lying under these spaces.

Resumen En este texto buscamos mostrar la necesidad de reconsiderar el espacio como categoría fundamental para evaluar lo común y la violencia que actualmente se ejerce contra éste. La consideración sobre el espacio surge históricamente del análisis filosófico del lenguaje en el siglo XX. A partir de ahí, sin embargo, se perfila un pensamiento más general sobre la estructura y, de ahí, sobre la espacialidad, que debe alcanzar la misma dignidad filosófica que el tiempo. Sobre esta base, se avanza sobre la estructura a la que apunta lo común, leído primero desde el pensamiento, luego desde el lenguaje, y finalmente a partir del espacio. Avanzamos la tesis de que el ser exige un espacio común, pensable a partir de una topología. Sentadas las bases de este espacio, se ofrecen algunas claves para pensar el espacio común contemporáneo y las violencias que hoy nos conciernen.

Abstract In the present article we reconsider space as a fundamental category to assess both “the common” and the actual violence exerted against it. This approach stems historically from the modern point of view in 20th century’s language philosophy. Nevertheless, it serves as a point of departure for a general consideration over structure and space natures. Such an enterprise to vindicate the role of space in philosophy, so called, in order to reach the same dignification as time. We first address its cognitive dimension, then its linguistical structure and finally arrive to a proper “spatial reading”. We advance in the thesis that a human being requires a common space, imagined through a topology. At the final remarks, we offer some cues to think contemporary common space to shed light on different sorts of violence directed against it.

Resumen. Nuestro propósito es presentar una exposición elemental de un resultado clásico en topología general, que es una versión débil de un problema conocido como la conjetura del espacio normal de Moore. Con este fin, se definen y estudian algunas propiedades básicas de los espacios de Moore y se caracterizan aquellos que a su vez son segundo contables y de Lindelöf. Además, se enuncia la hipótesis del continuo y el Axioma de Martin, y se aplican para establecer la independencia del problema en cuestión. MSC2010: 03E35, 03E50, 03E65, 54E30, 54E35.

Abstract. Our purpose is to present an elementary exposition of a classical result in general topology which is a weak version of a problem known as the normal Moore space conjecture. With this aim we study some of the basic properties of Moore spaces and characterize those which are both Lindelof and second countable. We also make use of the continuum hypothesis along with Martin’s axiom to establish the result in question.

Abstract The notion of Λ-fuzzy set was introduced by M. E. EI-Shafei and A. Zakari in 2006 ((20)). We examine some basic properties of it and prove some characterization theorems for the same. The paper presents a new class of fuzzy sets called fuzzy Λγ-sets that includes the class of all fuzzy γ-open sets. It also introduces the notion of fuzzy Vγ-sets as the dual concept of fuzzy Λγ sets to study the spaces constituted by those sets and obtain a completely different structure which is called fuzzy independent Alexandorff space. A stronger form of fuzzy Λb - continuity ((2)) called fuzzy Λγ-continuity is introduced and the relationships are also established with the already existing functions accordingly. Finally, fuzzy Λγ-Generalized closed sets are defined and studied with some of their applications.

Abstract In this paper a new class of sets termed as ωμ-regular open sets has been introduced and some of its properties are studied. We have introduced μ-nearly Lindelöfness in μ-spaces. We have shown that under certain conditions a μ-Lindelöf space [7] is equivalent to a μ-nearly Lindelöf space. Some properties of such spaces and some characterizations of such spaces in terms of ωμ -regular open sets are given.

Resumen El presente artículo propone un nuevo enfoque de clasificación de imágenes que utiliza como representación del espacio complejos cartesianos libre de ambigüedad en las relaciones topológicas. El enfoque propuesto comprende seis fases: (i) conversión de la imagen convencional al espacio de complejos cartesianos; (ii) transformación a niveles de grises; (iii) producción de super-píxeles basada en la transformada de cuenca; (iv) producción del espacio de textura aprovechando elementos inter-pixel 1-dimensionales; (v) clasificación mediante máquinas de soporte vectorial; y (vi) evaluación de los resultados. Aunque la exactitud temática de la clasificación a partir del nuevo enfoque mejora la exactitud obtenida con una representación convencional del espacio, la prueba de confianza indica que esa mejoría no es estadísticamente significativa. Sin embargo, el nuevo enfoque puede ser fortalecido en el futuro mediante la incorporación de técnicas para mejorar la definición de límites entre cuencas utilizando valores de probabilidad y espacios de textura multiescala.

Abstract This article proposes a new approach to image classification using a space representation as a Cartesian complex free of ambiguity in the topological relationships of adjacency, connectivity, and boundary. The proposed model comprises six phases: (i) image conversion from the conventional space into the Cartesian complex space; (ii) greyscale transformation, (iii) super-pixel space production based on watershed transform, (iv) texture space production taking advantage of 1-dimensional interpixel elements, (v) classification using support vector machines and (vi) results quality assessment. Although global accuracy of the proposed classification improves accuracy of results obtained with a conventional representation of space, the confidence test shows that this improvement is not statistically significant. However, the new approach can be further strengthened by incorporating techniques to improve boundaries definition between watersheds based on probability values as well as using spaces of multiscale texture.

Resumen En el presente trabajo, estudiamos los espacios de Brown, que son conexos y no completamente de Hausdorff. Utilizando progresiones aritméticas, construimos una base BG para una topología τG de N, y mostramos que (N, τG), llamado el espacio de Golomb, es de Brown. También probamos que hay elementos de BG que son de Brown, mientras que otros están totalmente separados. Escribimos algunas consecuencias de este resultado. Por ejemplo, (N, τG) no es conexo en pequeño en ninguno de sus puntos. Esto generaliza un resultado probado por Kirch en 1969. También damos una prueba más simple de un resultado presentado por Szczuka en 2010. MSC2010: 54D05, 11B25, 54D10, 54A05, 11B05, 11A07, 11A41.

Abstract In the present paper we study Brown spaces which are connected and not completely Hausdorff. Using arithmetic progressions, we construct a base BG for a topology τG on N, and show that (N, τG), called the Golomb space is a Brown space. We also show that some elements of BG are Brown spaces, while others are totally separated. We write some consequences of such result. For example, the space (N, τG) is not connected “im kleinen” at each of its points. This generalizes a result proved by Kirch in 1969. We also present a simpler proof of a result given by Szczuka in 2010.

Las siguientes son las notas de un mini curso que dí durante la escuela de verano CIMPA en Villa de Leyva, Colombia, en julio de 2014. El tema fue el trabajo que en conjunto se desarrolló con Damien Gayet sobre la topología de las hipersuperficies reales aleatorias, restringiéndonos al caso de los espacios proyectivos y enfocándonos en nuestras estimaciones inferiores. Particularmente, estimamos (por arriba y) por abajo la esperanza matemática de todos los números de Betti de las hipersuperficies reales proyectivas aleatorias de grado d. De hecho, para cualquier hipersuperficie cerrada y conexa Σ de Rn, estimamos por abajo la esperanza del número de componentes conexas de éstas hipersuperficies reales proyectivas aleatorias de grado d, las cuales son difeomorfas a Σ.

These are notes of the mini-course I gave during the CIMPA summer school at Villa de Leyva, Colombia, in July 2014. The subject was my joint work with Damien Gayet on the topology of random real hypersurfaces, restricting myself to the case of projective spaces and focusing on our lower estimates. Namely, we estimate from (above and) below the mathematical expectation of all Betti numbers of degree d random real projective hypersurfaces. For any closed connected hypersurface Σ of Rn, we actually estimate from below the mathematical expectation of the number of connected components of these degree d random real projective hypersurfaces which are diffeomorphic to Σ.