We introduce and study a new class of spaces, namely $beta-$topological vector spaces via $beta-$open sets. The relationships among these spaces with some existing spaces are investigated. In addition, some important and useful characterizations of $beta-$topological vector spaces are provided.

In this paper, we have dened and studied a generalized form of topological vector spaces called s-topological vector spaces. s-topological vector spaces are dened by using semi-open sets and semi-continuity in the sense of Levine. Along with other results, it is proved that every s-topological vector space is generalized homogeneous space. Every open subspace of an s-topological vector space is an s-topological vector space. A homomorphism between s-topological vector spaces is semi-continuous if it is s-continuous at the identity.

The notion of generalized locally bounded I-topological vector spaces is introduced. Some of their important properties are studied. The relationship between this kind of spaces and the locally bounded I-topological vector spaces introduced by Wu and Fang [Boundedness and locally bounded fuzzy topological vector spaces, Fuzzy Math. 5 (4) (1985) 87􀀀94] is discussed. Moreover, we also use the family of generalized fuzzy quasi-norms to char- acterize the generalized locally bounded I-topological vector spaces, and give some applications of this characterization.

In this paper, hyper vector spaces, in the sense of Krasner have been studied andVector (pseudo) fuzzy topological hypervector spaces have been introduced, as a generalization of them. In a fuzzy (quasi-)topological hypervector space, we show every its quotient hypervector space by a sub-hypervector space, is again a fuzzy (quasi-)topological hypervector space, and we will discuss its properties.Finally, we introduce the product of two fuzzy quasi-topological hypervector spaces on a specific hyperfield, and show that the product of two fuzzy quasi-topological hypervector spaces is a fuzzy quasi-topological hypervector spaces.Finally, we obtain some important properties of these topological fuzzy topological hypervector spaces.In this paper, hyper vector spaces, in the sense of Krasner have been studied andVector (pseudo) fuzzy topological hypervector spaces have been introduced, as a generalization of them. In a fuzzy (quasi-)topological hypervector space, we show every its quotient hypervector space by a sub-hypervector space, is again a fuzzy (quasi-)topological hypervector space, and we will discuss its properties.Finally, we introduce the product of two fuzzy quasi-topological hypervector spaces on a specific hyperfield, and show that the product of two fuzzy quasi-topological hypervector spaces is a fuzzy quasi-topological hypervector spaces.Finally, we obtain some important properties of these topological fuzzy topological hypervector spaces.

In this paper, we consider a hypervector space (in the sense of Tallini) $V$ over a field $K$. We use the fundamental relation $\varepsilon^*$ over $V$, as the smallest equivalence relation on $V$, to derived the fundamental vector space ${V}/\varepsilon^*$. In this regards, we prove that if $V$ is a (resp. quasi) topological hypervector space, then the fundamental vector space ${V}/\varepsilon^*$ with the property that each open subset of it is a complete part, then its fundamental vector space ${V}/\varepsilon^*$ is a topological vector space. Finally, we prove that for a topological vector space $(V,+,\cdot,K)$ and every subspace $W$ of $V$, the hypervector space $(\overline{V},+,\circ,K)$ is a topological hypervector space and we will prove $\overline{V}/\varepsilon^*$ and $V/W$ are homeomorphic, where $\overline{V}=V$.

Abstract. In this paper, the concept of local base with stratified structure in I-topological vector spaces is introduced. We prove that every I-topological vector space has a balanced local base with stratified structure. Furthermore, a new characterization of I-topological vector spaces by means of the local base with stratified structure is given.

Let X be a Hausdorff topological vector space over the field of real or complex numbers. When Vietoris topology is given, the hyperspace C(X ) of all nonempty compact subsets of X forms a topological exponential vector space over the same field. Exponential vector space [shortly, evs] is an algebraic ordered extension of vector space in the sense that every evs contains a vector space, and conversely, every vector space can be embedded into such a structure. A semigroup structure, a scalar multiplication and a partial order with some compatible topology comprise the topological evs structure. In this study, we have shown that besides C(X ), there are other hyperspaces namely P(X ), PBal(X ) PCV (X ), PNθ (X ), PS(X ), Pθ(X ) which have the same structure. To characterise the hyperspaces P(X ), C(X ) in light of evs, we have introduced some properties of evs which remain invariant under order-isomorphism. We have also introduced the concept of primitive function of an evs, which plays an important role in such characterisation. Lastly, with the help of these properties, we have characterised C(X ) as well as P(X ) as exponential vector spaces.

In this paper, first a brief history of equilibrium problems(EP) and generalized implicit vector equilibrium problems(GIVEP) are given. Then some existence theorems for GIVEP are presented, also some suitable conditions in order the solution set of GIVEP is compact and convex for set-valued mappings whose are a subset of the cartesian product of Hausdorff topological vector space and their range is a subset of a topological space values (not necessarily locally convex or a topological vector space). In almost all of published results for GIVEP the set-valued mappings are considered from a topological vector space(locally convex topological vector space) to a topological vector space while in this paper the range of the set-valued mappings are a subsets of a topological spaces. As applications of our results, we derive some suitable conditions for existing a normalized Nash equilibrium problems when the number of players are finite and the abstract case, that is infinite players. Finally, a numerical result, as an application of the main results, is given. The method used for proving the existence theorems is based on finite intersection theorems and Ky-Fan’s theorem. The results of this paper, can be considered as suitable generalizations of the published paper in this area.

The aim of this paper is to introduce and study the notion type of fibrewise topological spaces, namely fibrewise fuzzy j-topological spaces, Also, we introduce the concepts of fibrewise j-closed fuzzy topological spaces, fibrewise j-open fuzzy topological spaces, fibrewise locally sliceable fuzzy j-topological spaces and fibrewise locally sectionable fuzzy j-topological spaces. Furthermore, we state and prove several Theorems concerning these concepts, where $j={delta, theta, alpha, p, s, b, beta}$.

In this paper, we formalize the Menger probabilistic normed space as a category in which its objects are the Menger probabilistic normed spaces and its morphisms are fuzzy continuous operators. Then, we show that the category of probabilistic normed spaces is isomorphicly a subcategory of the category of topological vector spaces. So, we can easily apply the results of topological vector spaces in probabilistic normed spaces.