In this article, we have introduced the notion of pairwise COMPACTNESS in bi-isotonic spaces via finite intersection property and pairwise open cover. Moreover, we have given pairwise COMPACTNESS of bi-isotonic subspaces with both reduced closure and interior functions. We have characterized pairwise COMPACTNESS by the neighborhood concept; however, the axioms of bi-isotonic spaces are insufficient to prove the theorem, and we have studied this in bi-closure spaces. For similar reasons, occasionally, bi-closure spaces have been considered with additional explanations, even if some concepts related to COMPACTNESS have been naturally extended to bi-isotonic spaces. Additionally, interesting results have been obtained considering the pairwise Hausdorffness and COMPACTNESS relationship. It has also been observed that resembling cross-relationships exist for closed subsets of pairwise compact spaces. Finally, it has been observed that the pairwise COMPACTNESS of bi-closure spaces is preserved under bi-continuity.

One of the nice properties of the first-order logic is the COMPACTNESS of satisfiability. It states that a finitely satisfiable theory is satisfiable. However, different degrees of satisfiability in fuzzy logics will pose various kinds of COMPACTNESS in these logics.In this article, after an overview on the results around the COMPACTNESS of satisfiability and COMPACTNESS of $K$-satisfiability in H\'{a}jek Basic logic, some new results are given around this issue.It will be shown that there are topologies on $[0,1]$ and $[0,1]^2$ for which the interpretation of all logical connectives of the Basic logic is continuous. Furthermore, a topology on first-order structures will be introduced for any similarity relation as well. Then by the same ideas as in continuous logic, the results around the COMPACTNESS of satisfiability will be extended for Basic logic.

In this article, we use two concepts, measure of non-COMPACTNESS and Meir-Keeler condensing operators. The measure of non-COMPACTNESS has been applied for existence of solution nonlinear integral equations, ordinary differential equations and system of differential equations in the case of finite and infinite dimensions by some authors. Also Meir-Keeler condensing operators are shown in some papers such as [7, 11]. By the above concepts we can generalize some theorems which was proved by other authors, especially Darbo’s fixed point theorem. The space of our solution contains of all convergence sequences with a finite limit, such that with suitable norm is a Banach space. To reach our subject, we prove some theorems by using measure of non-COMPACTNESS and Meir-Keeler condensing operators to generalize some theorems which are created by some authors. For validity and application of our proposed theorems, we prove existence of solution for infinite system of second order differential equations with boundary conditions.

In this paper, we present the notion of weakly compact topo-logical fuzzes and give some characterizations of them. In particular, a characterization of weakly COMPACTNESS are given by the closedness of the projection fuzz maps. Also, we study some properties of proper fuzz maps as an important class of closed fuzz maps.

In this paper, the notion of almost S*-COMPACTNESS in L-topological spaces is introduced following Shi's definition of S*-COMPACTNESS. The properties of this notion are studied and the relationship between it and other definitions of almost COMPACTNESS are discussed. Several characterizations of almost S*-COMPACTNESS are also presented.

It is well known that, for Hausdorff spaces, a quotient map f : X -> Y is product stable, in the sense that the product map f x g is a quotient map, for every quotient map g, if and only if Y is locally compact and f is biquotient. Here, we present a generalization of those quotient maps which are product stable. 

In this paper, using pre-semi-open L-sets and their inequality, a new notion of PS-COMPACTNESS is introduced in L-topological spaces, where L is a complete De Morgan algebra. This notion does not depend on the structure of the basis lattice Land L does hot need any distributivity.

Some criteria for approximative COMPACTNESS of every weakly* closed convex set of Orlicz function spaces equipped with the Luxemburg norm are given. Although, criteria for approximative COMPACTNESS of Orlicz function spaces equipped with the Luxemburg norm were known, we can easily deduce them from our main results.

In this paper, the concept of countably near PS-COMPACTNESS in L-topological spaces is introduced, where L is a completely distributive lattice with an order-reversing involution. Countably near PS-COMPACTNESS is defined for arbitrary L-subsets and some of its fundamental properties are studied.