This note studies the relationship between Hutton's quasi-uniformities and Shi's quasi-uniformities. It is shown that when $L$ satisfies``multiple choice principle" for co-prime elements, the category of Hutton's quasi-uniform spaces is a bireflective full subcategory of the category of Shi's quasi-uniform spaces. Especially, if the remote-neighborhood mapping defined by Shi preserves arbitrary...

In this article, we establish sufficient conditions for the solution sets of parametric generalized quasi-variational relation problems with the stability properties such as the upper semicontinuity, lower semi-continuity, the Hausdorff lower semicontinuity, continuity, Hausdorff continuity, and closedness. Our results improve recent existing ones in the literature. Mathematics Subject Classifi...

Let R be a quasi-order on a compact Hausdorff topological space X. We prove that if X is scattered, then R satisfies the Priestley separation axiom if and only if R is closed in the product space X × X. Furthermore, if X is not scattered, then we show that there is a quasi-order on X that is closed in X × X but does not satisfy the Priestley separation axiom. As a result, we obtain a new charac...

In this paper, we introduce and study a class of generalized nonlinear vector quasi-variationallike inequalities with set-valued mappings in Hausdorff topological vector spaces which includes generalized nonlinear mixed variational-like inequalities, generalized vector quasi-variational-like inequalities, generalizedmixed quasi-variational-like inequalities and so on. Bymeans of fixed point the...

The classical theorems of Riesz [l] on compact operators have been extended by Leray [2] and Williamson [3] to the context of topological linear spaces. Ringrose [4] has shown that if an operator on such a space is compact, the square of its adjoint is also compact, where the topology on the dual space is that of uniform convergence on bounded sets. Thus if an operator is continuous and some po...

In this paper, we introduce and study the system of generalized vector quasi-variational-like inequalities in Hausdorff topological vector spaces. By means of fixed point theorem, we obtain existence theorem for the system of generalized vector quasi-variational-like inequalities and also derive the existence result of solutions for the generalized vector quasi-variational-like inequalities. Ke...

We investigate some basic descriptive set theory for countably based completely quasi-metrizable topological spaces, which we refer to as quasi-Polish spaces. These spaces naturally generalize much of the classical descriptive set theory of Polish spaces to the non-Hausdorff setting. We show that a subspace of a quasi-Polish space is quasi-Polish if and only if it is Π2 in the Borel hierarchy. ...