Supra b Limit Points and Supra b Separation Axioms

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 [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.

According to its definition, a topological space could be a highly unexpected object. There are spaces (indiscreet space) which have only two open sets: the empty set and the entire space. In a discrete space, on the other hand, each set is open. These two artificial extremes are very rarely seen in actual practice. Most spaces in geometry and analysis fall somewhere between these two types of spaces. Accordingly, the separation axioms allow us to say with confidence whether a topological space contains a sufficient number of open sets to meet our needs. To this end, we use bipolar hypersoft (BHS) sets (one of the efficient tools to deal with ambiguity and vagueness) to define a new kind of separation axioms called BHS Ti-space (i = 0, 1, 2, 3, 4). We show that ee BHS Ti-space (i = 1,2) implies BHS Ti−1-space; however, the converse is false, as shown by an example. e For i = 0, 1, 2, 3, 4, we prove that BHS Ti -space is hypersoft (HS) Ti -space and we present a condition so that eee HS Ti-space is BHS Ti-space. Moreover, we study that a BHS subspace of a BHS Ti-space is a BHS Ti-space for i = 0,1,2,3.

In this article, we introduced the concept of semi-weakly- (-sets based on weakly- open sets to obtain some semi-weakly- separation axioms. From theseIn this article, we introduced the concept of semi-weakly- (-sets based on weakly- open sets to obtain some semi-weakly- separation axioms. From these new concepts of separation axioms, we obtained axioms of semi-weakly-, semi-weakly-, and semi-weakly-spaces. We also derived some characterizations new concepts of separation axioms, we obtained axioms of semi-weakly-, semi-weakly-, and semi-weakly-spaces. We also derived some characterizations

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.

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.

This manuscript presents a new weaker version of supra septarian axioms based on supra $\epsilon$-open sets, along with its essential features, which are called supra-$\epsilon$-$T_{j}$-space, $j=0,1,2$, in the framework of supra topological spaces (or STSs). We give comprehensive explanations of each type of them, backed up by several examples and counterexamples that highlight the significance of our original approaches. We also provide a diagram that outlines these relationships. Additionally, we present the supra $\epsilon$-symmetric property and supra difference property and study their effects on these version of supra-$\epsilon$-septarian axioms. In especial, we show that the two concepts of supra-$\epsilon$-$T_{0}$-space and supra-$\epsilon$-$T_{1}$-space are the same for any STS that fulfills supra $\epsilon$-symmetric property. Finally, we study the supra topological and supra hereditary properties for each of the previously discussed approaches. In particular, we show that the property of being a supra-$\epsilon$-$T_{j}$-space, where $j=0,1,2$, is a supra-hereditary (topological) property.

We define the ??-closure operator as a new topological operator. We show that ??-closure of a subset of a topological space is strictly between its usual closure and its ?-closure. Moreover, we give several sufficient conditions for the equivalence between ??-closure and usual closure operators, and between ??-closure and ?-closure operators. Also, we use the ??-closure operator to introduce ??-open sets as a new class of sets and we prove that this class of sets lies strictly between the class of open sets and the class of ?-open sets. We investigate ??-open sets, in particular, we obtain a product theorem and several mapping theorems. Moreover, we introduce ?-T2 as a new separation axiom by utilizing ?-open sets, we prove that the class of !-T2 is strictly between the class of T2 topological spaces and the class of T1 topological spaces. We study relationship between ?-T2 and ?-regularity. As main results of this paper, we give a characterization of ?-T2 via ??-closure and we give characterizations of ?-regularity via ??-closure and via ??-open sets.

In this paper, we introduce the concept of N-closed maps and we obtain the basic properties and their relationships with other forms of N-closed maps in supra topological spaces.

A subset A of a topological space X is said to be: semi-open, if there exists an open set U such that U c A ~ U [8], semi-closed, if X \ A is semi-open [2, 3]. The union of any family of semi-open sets is semi-open [8]. The union of all semi-open sets which are contained in A is called the semi-interior of A; we denote it by s-Int A [2, 3]. From the definition it immediately follows that each semi-open (semi-closed) set has the Baire property. Let F be a multivalued map which assigns a non-empty subset F(x) of a topological space Y to each point xCX (for simplicity we will write F: X~Y). For any sets AcJ(, B c Y wewilldenote [I]: F(A)= U {F(x): xEA}, F+(B)={xEX:F(x)c cB}, F-(B)={xEX: e(x)NBr A multivalued map F: X ~ Y is said to be: upper (lower) quasi-continuous at a point XoEX, if for each open set V c Y such that F(Xo)c V (resp. F(xo)N V# O) and for each neighbourhood U of x0 there exists an open non-empty set U1cU such that F(x)cV (resp. F(x)NVr for each xC U1 [10, 12[, quasi-continuous at xoEX, if for any open sets V~, V~c Y such that F(xo)C c V1 and F(Xo) N 112 r ~ and for each neighbourhood U of x0 there exists an open non-emptyset U ~ U such that F(x)cV~ and F(x)NV2r for xEU~ [13]. Any single valued map f : X-~Y can be considered as a multivalued map with values {f(x)}. In this case each of the above three definitions gives the definition of quasi-continuity in the sense of Kempisty [6]. In the sequel the symbol E,(F), El(F) and E,(F) will be used to denote the sets of all points at which a multivalued map F is upper quasi-continuous, lower quasicontinuous or quasi-continuous respectively. It follows from the definitions that E,(F) c E u(F) NEt (F); the equality does not hold in general [4]. A multivalued map F is called upper quasi-continuous (lower quasi-continuous, quasi-continuous) if it has this property at each point Equivalently F is upper (lower) quasi-continuous iff for each open set V c Y the set F+(V) (resp. F-(V)) is semiopen. Similarly Fis quasi-continuous ifffor any open sets V~, V2~ Y the set F+(VI) N 0 F-(V2) is semi-open [13]. Let J( be a topological space and let Tq = {U',,,H: U is open, H is of the first category}. Then Tq is a topology on X [5] which sometimes is called the qualitative topology.

The significance fore supra topological spaces as a subject of study cannot be overstated, as they represent a broader framework than traditional topological spaces. Numerous scholars have proposed extension to supra open sets, including supra semi open sets, supra per open and others. In this research, a notion for ⱨ-supra open created within the generalizations of the supra topology of sets. Our investigation involves harnessing this style of sets to introduce modern notions in these spaces, specifically supra ⱨ - interior, supra ⱨ - closure, supra ⱨ - limit points, supra ⱨ - boundary points and supra ⱨ - exterior of sets. It has been examining the relationship with supra open. The research was also enriched with many of characteristics of each concept. Building upon this set classification, we introduced several kinds of maps like supra ⱨ - continuous, supra ⱨ - open, supra ⱨ - tentative, supra ⱨ -globally and supra ⱨ - homeomorphism. Additionally, we have proven a collection of useful relationships for the aforementioned of functions. Furthermore, the research was enhanced with illustrative and refuting examples.

In this article, we introduce the notion of the complementary soft neighborhood and present three kinds of covering soft rough set (mathcal {CSR}) models. The basic properties of these models are investigated. The relationships among these models are also discussed. Moreover, we establish the topological approach to mathcal {CSR} say, varDelta -topological spaces (i.e., varDelta -TS). Hence, the topological properties for varDelta -TS models such as varDelta -open sets, varDelta -closed sets, varDelta -interior, varDelta -closure, varDelta -boundary, varDelta -neighborhood and varDelta -limit point are studied and the relationships between them are given. Finally, we make use of an algorithm for these proposed models to deal with uncertainties for solving the MGDM problems using the constructed topologies.

Summary Józef Białas and Yatsuka Nakamura has completely formalized a proof of Urysohn’s lemma in the article [4], in the context of a topological space defined via open sets. In the Mizar Mathematical Library (MML), the topological space is defined in this way by Beata Padlewska and Agata Darmochwał in the article [18]. In [7] the topological space is defined via neighborhoods. It is well known that these definitions are equivalent [5, 6]. In the definitions, an abstract structure (i.e. the article [17, STRUCT 0] and its descendants, all of them directly or indirectly using Mizar structures [3]) have been used (see [10], [9]). The first topological definition is based on the Mizar structure TopStruct and the topological space defined via neighborhoods with the Mizar structure: FMT Space Str. To emphasize the notion of a neighborhood, we rename FMT TopSpace (topology from neighbourhoods) to NTopSpace (a neighborhood topological space). Using Mizar [2], we transport the Urysohn’s lemma from TopSpace to NTop-Space. In some cases, Mizar allows certain techniques for transporting proofs, definitions or theorems. Generally speaking, there is no such automatic translating. In Coq, Isabelle/HOL or homotopy type theory transport is also studied, sometimes with a more systematic aim [14], [21], [11], [12], [8], [19]. In [1], two co-existing Isabelle libraries: Isabelle/HOL and Isabelle/Mizar, have been aligned in a single foundation in the Isabelle logical framework. In the MML, they have been used since the beginning: reconsider, registration, cluster, others were later implemented [13]: identify. In some proofs, it is possible to define particular functors between different structures, mainly useful when results are already obtained in a given structure. This technique is used, for example, in [15] to define two functors MXR2MXF and MXF2MXF between Matrix of REAL and Matrix of F-Real and to transport the definition of the addition from one structure to the other: [...] A + B -> Matrix of REAL equals MXF2MXR ((MXR2MXF A) + (MXR2MXF B)) [...]. In this paper, first we align the necessary topological concepts. For the formalization, we were inspired by the works of Claude Wagschal [20]. It allows us to transport more naturally the Urysohn’s lemma ([4, URYSOHN3:20]) to the topological space defined via neighborhoods. Nakasho and Shidama have developed a solution to explore the notions introduced in various ways https://mimosa-project.github.io/mmlreference/current/ [16]. The definitions can be directly linked in the HTML version of the Mizar library (example: Urysohn’s lemma http://mizar.org/version/current/html/urysohn3.html#T20).

Weak Hausdorff separation axiom in I-fuzzy topological spaces and its application

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.