On almost and weakly i p -continuous functions in topological space.

English

A new class of closed sets, known as semi-regular weekly closed sets, is introduced in ideal topological space. In addition, we define a new class of continuous functions known as semi-regular weekly -continuous functions is introduced and studied. Some of their characteristics have also been examined and discussed interesting properties of semi-regular weekly -irresolute functions, strongly semi-regular weekly -continuous functions, and semi- regular weekly -continuous functions.

We introduce a new class of functions called strongly precontinuous functions in topological spaces. Some properties and several characterizations of this type of functions are obtained. We investigate the relationships between this class of functions and other classes of non-continuous functions.

In this paper, we introduce sufficient conditions for the non—empty intersection of two set—valued mappings in topological spaces. As applications, some topological minimax inequalities for two functions in which one of them is separately lower (or upper) semicontinuous are given. Finally, by employing our topological intersection theorems for two set—valued mappings, some other minimax inequalities have been derived without separately lower (or upper) sernicontinuity under but with another condition. These results are topological versions of corresponding minimax inequalities for two functions due to Fan (1964) and Sion (1958) in topological vector spaces

A n-dimensional topological manifold is defined as an n-dimensional local Euclidean space M that is also a second countable space and a Hausdorff space. If a topological space has a countable base, it is referred to as a second countable space. At the same time, continuous function and metric space can help people to understand topological space better. This article will express some basic ideas about continuous functions, metric space, topological space and its properties, and topological manifolds. Moreover, the paper shows some basic ideas of how topological manifolds, topological space and function could be recognized and proved. Through analysis, this paper demonstrates the connection between them, such as using properties of continuous function to prove the definition and properties of topological manifolds and spaces.

Cauchy-subregular functions vis-à-vis different types of continuity

In this paper, we provide the topological space of the iR-open set. Such collections are used to define and analyze the concepts of iR-continuous and iR-contra continuous functions and some of their properties. And preservation theorems about relationships between these functions and other functions in related classes are also explored. Also, on the other hand, we give examples to show that the convers may not be true

The goal of the research is to introduce new types of maps called semi totally Bc-continuous map and totally Bc-continuous map furthermore, study its properties. Additionally, we study the relationship of these functions and other known mappings are discussed.

We give some characterizations of rg-compact, gpr-compact and gpr-connected spaces by utilizing rg-open, gpr-open and gpr-closed sets. The paper is closely related to [A.M.Al-Shibani, rg-compact spaces and rg-connected spaces, Mathematica Pannonica, 17/1 (2006), 61-68], [Y.Gnanambal and K.Balachandran, On gpr-continuous functions in topological spaces,Indian .J.Pure appl.Math., 30(6) (1999),581-593] and [P.Gnanachandra et. al., Ultra Scientist, 24(1) A (2012), 185-191]

A. In this study, we investigate the concepts of -continuous, -irresolute, -open, and -closed mappings.We establish that every continuous mapping is inherently -irresolute and that every irresolute mapping is -continuous.However, the converse implications do not necessarily hold.This distinction sets -continuous mappings apart from traditional continuous mappings, particularly since the composition of two -continuous mappings may not always preserve -continuity.We propose several methods for constructing new -continuous (or -irresolute) mappings from existing ones, including pasting-type lemmas specifically tailored for these mappings.Additionally, we present counterexamples to illustrate and clarify these concepts.

We develop topological methods for the investigation of operator inclusions in Banach spaces, prove the generalized Ky Fan inequality, and study the critical points of many-valued mappings in topological spaces.

<p>In this paper, we introduce and studies a new class of closed and open maps is called sαrw-closed and sαrw-open maps in topological space. Also some of their properties have been investigated. We also introduce sαrw*-closed and sαrw*-open maps in topological spaces and study of their properties.</p>

A holomorphically fibred space generates locally trivial bundles with positive dimensional fibers. This paper proposes two varieties of fibrations (compact and non-compact) in the non-uniformly scalable quasinormed topological (C, R) space admitting cylindrically symmetric continuous functions. The projective base space is dense, containing a complex plane, and the corresponding surjective fiber projection on the base space can be fixed at any point on real subspace. The contact category fibers support multiple oriented singularities of piecewise continuous functions within the topological space. A composite algebraic operation comprised of continuous linear translation and arithmetic addition generates an associative magma in the non-compact fiber space. The finite translation is continuous on complex planar subspace under non-compact projection. Interestingly, the associative magma resists transforming into a monoid due to the non-commutativity of composite algebraic operation. However, an additive group algebraic structure can be admitted in the fiber space if the fibration is a non-compact variety. Moreover, the projection on base space supports additive group structure, if and only if the planar base space passes through the real origin of the topological (C, R) space. The topological analysis shows that outward deformation retraction is not admissible within the dense topological fiber space. The comparative analysis of the proposed fiber space with respect to Minkowski space and Seifert fiber space illustrates that the group algebraic structures in each fiber spaces are of different varieties. The proposed topological fiber bundles are rigid, preserving sigma-sections as compared to the fiber bundles on manifolds.

A new class of functions, called slightly generalized β-continuous functions is introduced.Basic properties of slightly generalized β-continuous functions are studied.The class of slightly generalized βcontinuous functions properly includes the class of slightly β-continuous functions and generalized β-continuous functions.Also, by using slightly generalized β-continuous functions, some properties of domain/range of functions are characterized.

A Universal Approximation Theorem for Gaussian-Gated Mixture of Experts Models