A NEW DIMENSION TYPE

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.

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.

Consider a dominance relation (a preorder) ≿ on a topological space X, such as the greater than or equal to relation on a function space or a stochastic dominance relation on a space of probability measures. Given a compact set K ⊆ X, we study when a continuous real function on K that is strictly monotonic with respect to ≿ can be extended to X without violating the continuity and monotonicity conditions. We show that such extensions exist for translation invariant dominance relations on a large class of topological vector spaces. Translation invariance or a vector structure are no longer needed when X is locally compact and second countable. In decision theoretic exercises, our extension theorems help construct monotonic utility functions on the universal space X starting from compact subsets. To illustrate, we prove several representation theorems for revealed or exogenously given preferences that are monotonic with respect to a dominance relation.

On the universal completion of pointfree function spaces

This work focuses on rough approximation operators generated from milli topological spaces and examines their features. For this purpose, we show how milli continuous functions are constructed. Further, exploiting the concepts of milli contra continuous, milli perfectly continuous, milli strongly continuous functions and milli basis in milli topological spaces, a real-life application is presented. The essential food habits for a healthy lifestyle were identified. Here the concepts of the topological reduction of attributes in a complete information system is used to get the key factors. A general algorithm for identifying the key factors is provided.

In this paper, we introduce two new classes of continuous functions in topological spaces called almost Ip-continuous and weakly Ip-continuous functions. We obtain several characterizations of them and some of their properties. Also, we investigate their relationships with some other types of functions.

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.

This chapter pursues three lines of investigation in the subject of functional analysis—one involving smooth functions and distributions, one involving fixed-point theorems, and one involving spectral theory.Section 1 introduces topological vector spaces. These are real or complex vector spaces with a Hausdorff topology in which addition and scalar multiplication are continuous. Examples include normed linear spaces, spaces given by a separating family of countably many seminorms, and weak and weak-star topologies in the context of Banach spaces. Various general properties of topological vector spaces are proved, and it is proved that the quotient of a topological vector space by a closed vector subspace is Hausdorff and is therefore a topological vector space.Section 2 introduces a topology on the space C∞(U) of smooth functions on an open subset of ℝN. The support of a continuous linear functional on C∞(U) is defined and shown to be a compact subset of U. Accordingly, the continuous linear functionals are called distributions of compact support.Section 3 studies weak and weak-star topologies in more detail. The main result is Alaoglu’s Theorem, which says that the closed unit ball in the weak-star topology on the dual of a normed linear space is compact. In an earlier chapter a preliminary form of this theorem was used to construct elements in a dual space as limits of weak-star convergent subsequences.Section 4 follows Alaoglu’s Theorem along a particular path, giving what amounts to a first example of the Gelfand theory of Banach algebras. The relevant theorem, known as the Stone Representation Theorem, says that conjugate-closed uniformly closed subalgebras containing the constants in B(S) are isomorphic via a norm-preserving algebra isomorphism to the space of all continuous functions on some compact Hausdorff space. The compact space in question is the space of multiplicative linear functionals on the subalgebra, and the proof of compactness uses Alaoglu’s Theorem.Sections 5–6 return to the lines of study toward distributions and fixed-point theorems. Section 5 studies the relationship between convexity and the existence of separating linear functionals. The main theorem makes use of the Hahn-Banach Theorem. Section 6 introduces locally convex topological vector spaces. Application of the basic separation theorem from the previous section shows the existence of many continuous linear functionals on such a space.Section 7 specializes to the line of study via smooth functions and distributions. The topic is the introduction of a certain locally convex topology on the space C ∞com (U) of smooth functions of compact support on U. This is best characterized by a universal mapping property introduced in the section.Sections 8–9 pursue locally convex spaces along the other line of study that split off in Section 5. Section 8 gives the Krein-Milman Theorem, which asserts the existence of a supply of extreme points for any nonempty compact convex set in a locally convex topological vector space. Section 9 relates compact convex sets to the subject of fixed-point theorems.Section 10 takes up the abstract theory of Banach algebras, with particular attention to commutative C* algebras with identity. Three examples are the algebras characterized by the Stone Representation Theorem, any L∞ space, and any adjoint-closed commutative Banach algebra consisting of bounded linear operators on a Hilbert space and containing the identity.Section 11 continues the investigation of the last of the examples in the previous section and derives the Spectral Theorem for bounded self-adjoint operators and certain related families of operators. Powerful applications follow from a functional calculus implied by the Spectral Theorem. The section concludes with remarks about the Spectral Theorem for unbounded self-adjoint operators.KeywordsBanach AlgebraTopological Vector SpaceNormed Linear SpaceInternal PointCompact Hausdorff SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

It is proved that if -V is a metric space, E a Banach space containing a a-weakly-compact dense subset, then the space (Mr(X, E'), o(MT(X, £"), Cb(X, E))) is angehe, Cb(X, E) being all bounded continuous functions from X into E and MT(X, £") the dual of Cb(X, E) with the strict topology ß.A Hausdorff topological space Y is called angelic if (i) every relatively countably compact subset of Y is relatively compact, and (ii) for any point x in the closure of a relatively compact subset A of Y, there exists a sequence, in A, converging to x [4, p. 534].In this paper X will stand for a completely regular Hausdorff space, E a Banach space over K, the field of real or complex numbers, C(X, E) (C(X)) all £-valued (AT-valued) continuous functions on X, and Cb(X, E) (Cb(X)) all bounded £-valued (/T-valued) continuous functions on X.We shall use the notations of [5] for locally convex spaces.Also the notations and results from [2] will be used.The topologies ß, ß0, ßx, ße (also denoted by ßx) are defined on Cb(X) in [9], [7],[8]-these topologies are defined for K = R, the reals, but naturally extend to K = C, the complex field, ß, ß0, ßx, ßx are defined on Cb(X, E) in [1], [2].If %°° = %X(X, E) = [H c Cb(X, E): H pointwise equicontinuous and uniformly bounded}, the topology ßx is the finest locally convex topology on Cb(X, E) agreeing with pointwise topology on each H E %°°.It is well known that MT(X, E') = (Cb(X, E), ß)', M,(X, E') = (Cb(X, E), ß0)' and MX(X, E') = (Cb(X, E), ßj (see [2]).If J is a metric, MX(X, E') = Mr(X, £") and ß and ßx are both Mackey [2]; from this it follows that ß = ßx in this case.If X is a metric space it is proved in [5, Theorem 5.3] that (Moe(X), o(Mao(X), Cb(X))) is angelic.We will extend this result to MX(X, E') for some special Banach spaces E.Theorem 1. (F', t(F', F)) is complete, t denoting the Mackey topology of the duality <F, F'), where F = Cb(X, E) and F' = MX(X, E').Proof.By Grothendieck completeness theorem [5, Theorem 6.2, p. 148], it is enough to prove that any linear form p on F which is continuous on every

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.

The lattice A1(U ∨(X)) of subalgebras with unit of the semifield U ∨(X) of continuous positive functions defined on a topological space X is considered. A topological space is said to be a Hewitt space if it is homeomorphic to a closed subspace of a Tychonoff power of the real line R. The main achievement of the paper is the proof of the fact that any Hewitt space X is determined by the lattice A1(U ∨(X)).

Chapter 12 N-Compactness and Its Applications

In this paper we desire to extend the neutrosophic topological spaces into N-neutrosophic topological spaces. Also we show that this theory can be deduced toN-intuitionistic and N-fuzzy topological spaces etc. Further we develop not only the concept of classical generalized closed sets into N-neutrosophic topological spaces but also obtain its basic properties. Finally we investigate its continuous function and generalized continuous function.

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.

This work is devoted to the study of kernel lattices of semifields of continuous positive functions defined on some topological space. It is established that they are lattices with a pseudo-complement. New characterizations of the following properties of topological spaces are obtained in terms of kernels, predominantly principal kernels, and semifields of continuous functions: to be an F-space, to be a P-space, basic and extremal disconnectedness, pseudo-compactness, and finiteness.