The focus of this paper is to employ a new topological condition, based on admissibility of the function space topology, to provide existence results for a generalized vector quasi-variational inequality problem and its stronger form as well. The inequality problems involve a set-valued function, and the existence results are proved without using any monotonicity or convexity assumptions on the functions. Further, the solution sets of the inequality problems are shown to be closed and compact.

In this paper, we are devoted to investigating the existence of the Generalized Vector Quasi-Variational-Like Inequality Problem by using a new approach. Here we use admissibility of the function space topology and convergence of nets to establish the required existence results. We also introduce the concept of adic maps. Moreover, we provide an example to show that our solutions are independent of the solutions available in the literature.

Dual topology for the function space topologies for multifunctions are introduced and investi-gated. It is found that a topology T on CM(Y, Z) is splitting (resp. admissible) if and only if its dual pair (T+, T-) is splitting (resp. admissible). Similarly, the pair (T+, T-) is splitting (resp. admissible) if and only if its dual T(T+,T-) is splitting (resp. admissible).

Using the concept of m-open sets, M-regularity and M-normality are introduced and investigated. Both these notions are closed under arbitrary product. M-normal spaces are found to satisfy a result similar to Urysohn lemma. It is shown that closed sets can be separated by m-continuous functions in a regular space.

The notion of dual uniformity is introduced on UC(Y,Z)UC(Y,Z), the uniform space of uniformly continuous mappings between YY and ZZ, where (Y,V)(Y,{\mathcal{V}}) and (Z,U)(Z,{\mathcal{U}}) are two uniform spaces. It is shown that a function space uniformity on UC(Y,Z)UC(Y,Z) is admissible (resp. splitting) if and only if its dual uniformity on UZ(Y)={f2−1(U)∣f∈UC(Y,Z),U∈U}{{\mathcal{U}}}_{Z}(Y)=\{{f}_{2}^{-1}(U)\hspace{0.33em}| \hspace{0.33em}f\in UC(Y,Z),U\in {\mathcal{U}}\} is admissible (resp. splitting). It is also shown that a uniformity on UZ(Y){{\mathcal{U}}}_{Z}(Y) is admissible (resp. splitting) if and only if its dual uniformity on UC(Y,Z)UC(Y,Z) is admissible (resp. splitting). Using duality theorems, it is also proved that the greatest splitting uniformity and the greatest splitting family open uniformity exist on UZ(Y){{\mathcal{U}}}_{Z}(Y) and UC(Y,Z)UC(Y,Z), respectively, and these two uniformities are mutually dual splitting uniformities.

In this paper, we discuss several variants of the η-generalized vector variational-like inequality problem and provide existence theorems for their solutions via a topological approach. Several topological concepts like compactness, closedness, net theory and admissibility of function space topology are used for obtaining the main results. Finally, we give some topological properties of the solution set so obtained.

A uniform study towards normality is provided for topological spaces. Following Császár, $\gamma$-normality and $\gamma$($\theta$)-normality are introduced and investigated. For $\gamma \in \Gamma_{13}$, $\gamma$-normality is found to satisfy Urysohn's lemma and provide partition of unity. Several existing variants of normality such as $\theta$-normality, $\Delta$-normality etc. are shown to be particular cases of $\gamma$($\theta$)-normality. In this process, $\gamma$-regularity and $\gamma$($\theta$)-regularity are introduced and studied. Several important characterizations of all these notions are provided.

In this paper, we discuss two variants of the generalized nonlinear vector variational-like inequality problem. We provide their solutions by adopting topological approach. Topological properties such as compactness, closedness, and net theory are used in the proof. The admissibility of the function space topology and KKM-Theorem have played important role in proving the results.

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 New families of uniformities are introduced on U C ( X , Y ) UC(X,Y) , the class of uniformly continuous mappings between X and Y, where ( X , U ) (X,{\mathcal{U}}) and ( Y , V ) (Y,{\mathcal{V}}) are uniform spaces. Admissibility and splittingness are introduced and investigated for such uniformities. Net theory is developed to provide characterizations of admissibility and splittingness of these spaces. It is shown that the point-entourage uniform space is splitting while the entourage-entourage uniform space is admissible.

If X is a topological space and A ⊆ X, then the number of distinct sets that can be obtained from A by using all possible compositions for operators iγ , cγ (where γ = σ, π, α, β) introduced by Császár is at the most 25.Explicit expressions for these sets are provided.An example is provided where all the 25 different sets are determined.The result is also discussed for special cases such as when the space is extremally disconnected, resolvable, open-unresolvable, and partition spaces.MSC (2010)

Function space topologies are investigated for the class of continuous multifunctions. Using the notion of continuous convergence, splittingness and admissibility are discussed for the topologies on continuous multifunctions. The theory of net of sets is further developed for this purpose. The (τ,μ)-topology on the class of continuous multifunctions is found to be upper admissible, while the compact-open topology is upper splitting. The point-open topology is the coarsest topology which is coordinately admissible, it is also the finest topology which is coordinately splitting.

$(\lambda, \mu)$-regularity and $(\lambda, \mu)$-normality are defined for generalized topological spaces. Several variants of normality existing in the literature turn out to be particular cases of $(\lambda, \mu)$-normality. Uryshon's lemma and Titze's extension theorem are discussed in the light of ($\lambda, \mu$)-normality.

We define and study a new class of regular sets calledPS-regular sets. Properties of these sets are investigated for topological spaces and generalized topological spaces. Decompositions of regular open sets and regular closed sets are provided usingPS-regular sets. Semiconnectedness is characterized by usingPS-regular sets.PS-continuity and almostPS-continuity are introduced and investigated.

Convergence theory for the generalized fuzzy topological spaces is developed. Some important subclasses of Γ(X), the class of monotonic mappings on X, are also discussed. 2010 AMS Classification: 54A20

Extremal disconnectedness is further investigated for generalized topological spaces. It is found that extremally disconnected generalized topological spaces are a rich source of generalized lower semi-continuous and generalized upper semi-continuous mappings.