$L$-Topological Spaces

The notion of fuzzy sets initiated to overcome the uncertainty of an object. Fuzzy topological space, in- tuitionistic fuzzy sets in topological structure space, vagueness in topological structure space, rough sets in topological space, theory of hesitancy and neutrosophic topological space, etc. are the extension of fuzzy sets. Soft set is a family of parameters which is also a set. Fuzzy soft topological space, intuitionistic fuzzy soft and neutrosophic soft topological space are obtained by incorporating soft sets with various topological structures. This motivates to write a review and study on various soft set concepts. This paper shows the detailed review of soft topological spaces in various sets like fuzzy, Intuitionistic fuzzy set and neutrosophy. Eventually, we compared some of the existing tools in the literature for easy understanding and exhibited their advantages and limitations.

In 2016 A. Devika and A. Thilagavathi introduced a new class of sets called M*-open sets and investigated some properties of these sets in topological spaces. In this paper, we introduce and study a new class of spaces, namely M*-irresolute topological vector spaces via M*-open sets. We explore and investigate several properties and characterizations of this new notion of M*-irresolute topological vector space. We give several characterizations of M*-Hausdorff space. Moreover, we show that the extreme point of the convex subset of M*-irresolute topological vector space X lies on the boundary.

We introduce and investigate the concepts of θ ω -limit points and θ ω -interior points, and we use them to introduce two new topological operators. For a subset B of a topological space Y , σ , denote the set of all limit points of B (resp. θ -limit points of B , θ ω -limit points of B , interior points of B , θ -interior points of B , and θ ω -interior points of B ) by D B (resp. D θ B , D θ ω B , Int B , Int θ B , and Int θ ω B ). Several results regarding the two new topological operators are given. In particular, we show that D θ ω B lies strictly between D B and D θ B and Int θ ω B lies strictly between Int θ B and Int B . We show that D B = D θ ω B (resp. Cl θ B = Cl θ ω B and D B = D θ ω B = D θ B ) for locally countable topological spaces (resp. antilocally countable topological spaces and regular topological spaces). In addition to these, we introduce several product theorems concerning metacompactness.

A topological space is called a uqu space [10] if it admits a unique quasi-uniformity. Answering a question [2, Problem B, p. 45] of P. Fletcher and W. F. Lindgren in the affirmative we show in [8] that a topological space X is a uqu space if and only if every interior-preserving open collection of X is finite. (Recall that a collection ℒ of open sets of a topological space is called interior-preserving if the intersection of an arbitrary subcollection of ℒ is open (see e.g. [2, p. 29]).) The main step in the proof of this result in [8] shows that a topological space in which each interior-preserving open collection is finite is a transitive space. (A topological space is called transitive (see e.g. [2, p. 130]) if its fine quasi-uniformity has a base consisting of transitive entourages.) In the first section of this note we prove that each hereditarily compact space is transitive. The result of [8] mentioned above is an immediate consequence of this fact, because, obviously, a topological space in which each interior-preserving open collection is finite is hereditarily compact; see e.g. [2, Theorem 2.36]. Our method of proof also shows that a space is transitive if its fine quasi-uniformity is quasi-pseudo-metrizable. We use this result to prove that the fine quasi-uniformity of a T1 space X is quasi-metrizable if and only if X is a quasi-metrizable space containing only finitely many nonisolated points. This result should be compared with Proposition 2.34 of [2], which says that the fine quasi-uniformity of a regular T1 space has a countable base if and only if it is a metrizable space with only finitely many nonisolated points (see e.g. [11] for related results on uniformities). Another by-product of our investigations is the result that each topological space with a countable network is transitive.

In [8], Császár introduced some generalized open sets in generalized topological spaces, namely, μ-semi-open, μ-preopen, μ-β-open, and μ-α-open. The primary purpose of this paper is to study some new properties and characterizations of these concepts as we also introduce and study a new generalized open set in generalized topological spaces, namely, μ-b-open. We also investigate some relationships between these concepts.

In the current study, the idea of S ψ φ -open sets is used to introduce some new types of nearly open sets in topological space, where ψ = { θ , δ , τ } and φ = { θ , δ , τ , α , a } , thereby illustrating some of their basic qualities and the links between them. Some characterizations, topological properties, mappings, and separation axioms of these new concepts are also presented.

In this work we introduce the supra iR-open set in supra topological spaces. , a novel class of supra i-open sets. This class of sets exactly lies between supra regular open and supra i- open sets. We also look at its basic characteristics and compare it to other sorts of data supra semi open sets types as supra i- open set, supra regular open, regular open, regular closed, clopen, supra semi regular open set and supra clopen . By using this set , We introduce and define the notion of supra totally iR-continuous , supra iR-irresolute map and investigate some of its properties. We prove that every supra totally iR- continuous function is supra iR-irresolute, every totally i????-continuous function is supra totally iR continuous and any strongly continuous function is supra totally iR-continuous, on the other hand, we give examples to show that the convers may not be true. At last , We have proved some theorem to discuss the properties of supra iR-normal space and supra iR-regular space.

In many papers, new classes of sets had been studied in topological space, then the notion of continuity between any two topological spaces (a function from X to Y is continuous if the inverse image of each open set of Y is open in X) is studied via this new classes of sets. Here the authors also introduce new classes of sets called pj-b-preopen, pj-b-B set, pj-b-t set, pj-b-semi-open and pj-sb-generalized closed set in bitopological space [1] which is a set with two topologies defined on it, then they study the notion of continuity via this set and introduce some of the theories which are studying the decomposition of continuity via this set in bitopological space.

In this paper, we introduce a new concept called semi-perfect sets in topological spaces. We define semi-isolated points and semi-perfect sets. Further, some interesting results and properties of these concepts are investigated.

This paper introduces the notion of generalized hyperconnected sets in topological spaces using Levine’s concept of generalized closed sets. We define generalized hyperconnected sets and generalized hyperconnected components and study their fundamental properties. Furthermore, we provide necessary conditions for a topological space to be generalized hyperconnected. The relationship between this type of connectedness and other types is also presented. Several results concerning these new concepts are obtained and proven.

A fixed‐point theorem is proved under noncompact setting of general topological spaces. By applying the fixed‐point theorem, several new existence theorems of solutions for equilibrium problems are proved under noncompact setting of topological spaces. These theorems improve and generalize the corresponding results in related literature.

The concept main of paper is to connected between two definitions pre-open sets and g-open sets in the topological spaces is called Hpre-open sets, study restudy relations them, also present a new definitions, namely Hpre-continous, Hpre-open functions, Hpre-closed functions and have proved some theorems about these concept. Finally, we have present the definitions and results of separation axioms by Hpre-open sets,namely Hpre - T1 spaces , Hpre - T2 spaces and studied some of its applications.

Existence of solutions for quasi-equilibrium problems in noncompact topological spaces

This paper explores the decomposition of pre--irresolute (pre--irr) maps and their connection to generalized closed sets (g-closed) in topological space (top. sp.). We provide the necessary background on top. sp., continuity, closed, and related concepts. We introduce the definitions of pre--irr maps and g-closed and discuss their basic properties. The decomposition theorem is presented, stating that every pre--irr map can be decomposed into a composition of a pre--continuous (pre- -cont.) map and a b-irr map. Examples and applications are provided to illustrate the behavior of pre--irr maps and the practical implications of the decomposition theorem. We examine the relationships between pre--cont. maps and pre--irr maps, exploring the conditions under which a pre--irr map is also -irr, and provide a proof for this relationship. The properties of pre--cont. maps, - maps, g- pre--cont. maps, and g--cont. maps are discussed, along with proofs and examples to illustrate their applicability. This paper contributes to the understanding of pre--irr maps, their decomposition, and their relationships with g-closed, providing a foundation for further research in top. sp. and map decompositions. Keywords: pre-β-irresolute maps, g-closed, pre-β-cont. maps, b-cont. maps

In this paper we use the following abbreviations: FB (filter base), MS (measurable space), AS (attraction set), DS (directed set), AD (attainability domain), GE (generalized element), s/s (subset), ApS (approximate solution), TS (topological space), and u/f (ultrafilter). We consider general questions connected with the asymptotic attainability; the latter is understood as the attainability of certain states under asymptotic constraints. One natural concrete variant of the problem which we consider below is related to the construction of the AD of a controlled system at a fixed time moment and the study of its properties. This concrete problem is not stable with respect to disturbances, in particular, for weakened constraints. One can treat weakened constraints (phase constraints, edge and intermediate ones) in this problem as asymptotic restrictions. Respectively, the limit of real AD which correspond to weakened constraints can be considered as an AS; from the practical point of view it plays the same role as a “regular ” AD does. Together with AD, with the strict and approximate fulfillment of constraints one can consider sheaves of trajectories of the controlled system and thus obtain sets in a functional space equipped with the corresponding topology. The latter can be generated by a metric, but this is not necessarily so (for example, in cases typical for control problems with continuous time, the topology of pointwise convergence is not metrizable). Therefore it is natural to consider logically similar problems on the attainability in TS, though this can require the application of more complex constructions for AS. This approach is used in the present paper. In addition, using such general statements of the problem, it is natural to assume asymptotic analogs of constraints which are not necessary connected with the weakening of some standard conditions. Really, weakening conditions of such a kind, we form a family of sets of feasible elements for each constraint weakened in a concrete way (generated by the set of mentioned conditions). From this moment we “deal” with the obtained family (really, this is a result of the relaxation procedure for the initial problem), using the minimal set of properties of sets that compose it. But this means that from the beginning we can assume that the family of s/s of the space of regular solutions (or controls) is given, not questioning how this family has been obtained (it is advisable to reduce the set of properties “to the minimum”). In particular, we do not tend to connect this family with a successive weakening of some “rigid ” conditions. In this case the question on the attainability in TS has an asymptotic character. Let us consider a very simple example of a static problem, fixing two finite-dimensional spaces R m and R n (m and n are natural numbers), and a bounded mapping f from R m to R n .F or de finiteness,