On ⱨ - supra open sets in supra topological spaces

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 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.

In this paper, we introduced study about a basic properties of adherent points by the class B*c – open. Also, we showed properties accumulation points by using B*c – open sets and we provided the relation among adherent points, β – adherent points and B*c – adherent points. Also, accumulation points, β – accumulation points and B*c – accumulation points.

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 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.

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.

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.

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.

In recent years, some generalized structures of topology were introduced. Supra topology was one the most important of those generalizations. To contribute in this orientation, we devoted this work to studying limit points and separation axioms on supra topological spaces by using supra $b$-open sets. We define them in a similar way of their counterparts on topological spaces. In general, we demonstrate their main properties and investigate the sufficient conditions for some equivalent relations between them. Some novel and interesting examples are provided.

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.

Many application scenarios for image recognition require learning of deep networks from small sample sizes in the order of a few hundred samples per class. Then, avoiding overfitting is critical. Common techniques to address overfitting are transfer learning, reduction of model complexity and artificial enrichment of the available data by, e.g., data augmentation. A key idea proposed in this paper is to incorporate additional samples into the training that do not belong to the classes of the target task. This can be accomplished by formulating the original classification task as an open set classification task. While the original closed set classification task is not altered at inference time, the recast as open set classification task enables the inclusion of additional data during training. Hence, the original closed set classification task is augmented with an open set task during training. We therefore call the proposed approach open set task augmentation. In order to integrate additional task-unrelated samples into the training, we employ the entropic open set loss originally proposed for open set classification tasks and also show that similar results can be obtained with a modified sum of squared errors loss function. Learning with the proposed approach benefits from the integration of additional “unknown” samples, which are often available, e.g., from open data sets, and can then be easily integrated into the learning process. We show that this open set task augmentation can improve model performance even when these additional samples are rather few or far from the domain of the target task. The proposed approach is demonstrated on two exemplary scenarios based on subsets of the ImageNet and Food-101 data sets as well as with several network architectures and two loss functions. We further shed light on the impact of the entropic open set loss on the internal representations formed by the networks. Open set task augmentation is particularly valuable when no additional data from the target classes are available—a scenario often faced in practice.

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).

In this paper we define a new type of connectedness by using b - open sets and discuss the relationship between this connectedness and various types of connectedness already defined in topological spaces.

Let X be a topological space, that is, a space with open sets such that the union of any collection of open sets is open and the intersection of any finite number of open sets is open. A covering of X is a collection of open sets whose union is X. The covering is called countable if it consists of a countable collection of open sets or finite if it consists of a finite collection of open sets ; it is called locally finite if every point of X is contained in some open set which meets only a finite number of sets of the covering. A covering is called a refinement of a covering U if every open set of X is contained in some open set of . The space X is called countably paracompact if every countable covering has a locally finite refinement.