On some properties of generalized open sets in generalized 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 neutro-fine topological space is a space that contains a combination of neutrosophic and fine sets. In this study, the various types of open sets such as generalized open and semi-open sets are defined in such space. The concept of interior and closure on semi-open sets are defined and some of their basic properties are stated. These definitions extend the concept to generalized semi-open sets. Moreover, the minimal and maximal open sets are defined and some of their properties are studied in this space. As well as, discussed the complement of all these sets as its closed sets. The basic properties of the union and intersection of these open sets are stated in some theorems. Only a few sets satisfy this postulates, and others are disproved as shown in the counterexamples. The converse of some theorems is proved in probable examples.

The aim of this paper is to introduce and study $\mathcal{B}$-open sets and related properties. Also, we define a bi-operator topological space $(X, \tau, T_1, T_2)$, involving the two operators $T_1$ and $T_2$, which are used to define $\mathcal{B}$-open sets. A $\mathcal{B}$-open set is, roughly speaking, a generalization of a $b$-open set, which is, in turn, a generalization of a pre-open set and a semi-open set. We introduce a number of concepts based on $\mathcal{B}$-open sets.

The aim of this paper is to introduce the concept of fuzzy soft tritopological Space, Where we define it as a space equipped with three fuzzy soft topologies, i.e. triple of three fuzzy soft topologies , and on the same initial universal set and the set of parameters .And investigate some of its fundamental properties.Also we defined some new kinds of fuzzy soft open sets in fuzzy soft tritopological spaces, which are called fuzzy soft -open set, fuzzy soft -pre-open set, fuzzy soft semi-open set, fuzzy soft --open set (fuzzy soft tri--open set) and fuzzy soft *open set.We can consider this work as an introduction of this concept.

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.

The main objective of this paper is to introduce a new generalized form of topological vector spaces, namely, almost $\\alpha$-topological vector spaces by using the concepts of $\\alpha$-open sets, regular open sets, and almost $\\alpha$-continuous mappings. In the present paper, some important examples of almost $\\alpha$-topological vector spaces with basic properties are characterized.

In this paper, a new kind of set called a \(\gamma\)-semi-open set is introduced and investigated using the \(\gamma\)-operator due to Ogata. Such sets are used for studying new types of mappings, viz. \(\gamma\)-semi-continuous, \((\gamma,\beta)\)-semi-irresolute and weakly \(\gamma\)-semi-continuous.

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 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 the present paper, we introduced a class of fuzzy G- semi open sets which is contained in class of fuzzy semi open sets. Using them fuzzy G- semi continuous functions and some special functions called fuzzy slightly G- semi continuous functions is introduced. Further it is shown in fuzzy subset theory, each fuzzy slightly G- semi continuous functions is fuzzy slightly semi continuous function.

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.

The concept of complement function is used to define a fuzzy closed subset of a fuzzy topological space. That is a fuzzy subset l is fuzzy closed if the standard complement 1-l = l¢ is fuzzy open. Here the standard complement is obtained by using the function C: [0, 1]® [0, 1] defined by C (x) = 1-x, for all x I[0, 1]. Several fuzzy topologists used this type of complement while extending the concepts in general topological spaces to fuzzy topological spaces. But there are other complements in the fuzzy literature. This motivated the second and third authors to introduce the concepts of fuzzy C -closed sets, and fuzzy C - pre closed sets, in fuzzy topological spaces. In this paper, we generalize the concept of fuzzy strongly pre open and fuzzy strongly pre closed sets by using the arbitrary complement function C, instead of the usual fuzzy complement function, and by using fuzzy C – pre closure instead of fuzzy pre closure. The concepts of fuzzy strongly C – pre open and strongly C – pre closed sets in fuzzy topological spaces and studied their properties in the third and fourth section, respectively. Key words: Fuzzy strongly C – pre open, fuzzy strongly C – pre closed sets and fuzzy topology.

The aim of this paper is to introduce the concepts of regularly continuous and fully continuous mappings as the mappings that have the preimages of regular open sets are regular open (respectively, open ). We investigate the connections between these classes and several well-known others of `generalized continuous mappings'. Several characterizations and decompositions of certain continuities are provided. In particular, a decomposition of complete continuity, which was first introduced by Arya and Gupta [1] , is also provided. Keywords: Regular open set, Preopen set, t-set, Continuous mapping.

We characterize codense and completely codense ideals. Introduction and preliminaries The subject of ideals in topological spaces has been studied by Kura- towski (10) and Vaidyanathaswamy (15). An ideal # on a topological space (X, ) is a collection of subsets of X which satisfies (i) A 2 # and B A implies B 2 # and (ii) A 2 # and B 2 # implies A(B 2 #. Given a topolog- ical space (X, ) with an ideal # on X and if }(X) is the set of all subsets of X, a set operator ( ) : }(X) ! }(X), called a local function (10) of A with respect to and # is defined as follows: for A X, A (#, ) = x 2 X | U \A 62 # for every U 2 @(x) where @(x) = {U 2 | x 2 U}. A Kuratowski closure operator cl ( ) for a topology (#, ), called the -topology, finer than is defined by cl (A) = A ( A (#, ) (15). When there is no chance for con- fusion, we will simply write A for A (#, ) and or (#) for (#, ). By a space, we always mean a topological space (X, ) with no separation properties assumed. If A X and # is an ideal on X, then cl(A) and int(A) will denote the closure and interior of A in (X, ), respectively, and cl (A) and int (A) will denote the closure and interior of A in (X, ), respectively. #n(X, ) or simply #n is the ideal of all nowhere dense subsets in (X, ). A subset A of a space (X, ) is said to be semiopen (13) if there is an open set G such that G A cl(G). The family of all semiopen sets in (X, ) is denoted by SO(X, ). Clearly, SO(X, ) and if A is semiopen and A 6 ;, then int(A) 6 ;. The complement of a semiopen set is semiclosed. A subset A of a space (X, ) is an -set (13) if A int(cl int(A) ). The family of all

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).