Baire sets in topological spaces

Chapter 12 N-Compactness and Its Applications

The Cambridge case for diversification planning: Towards stability in an R & D economy : Economic security committee, (Cambridge peace commission, 1986) pp. 16, $3.00

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.

0. Introduction For a topological space X, we denote by C(X) the set of all continuous real-valued functions on X, and by C*(X) the subset of C(X) composed of bounded functions. Both C(X) and C* (X) can be considered to be semigroups if the semigroup operation is the pointwise multiplication of functions. A classical theorem [6] states that, if X1, 2"2 are compact Hausdorff spaces, then the isomorphy of the semigroups C(2" 0 and C(X~) implies the homeomorphy of X~ and X~. The paper [8] contains the statement that the same holds if X~ and X 2 are realcompact spaces. It is easy to deduce from these statements two slightly more general ones in which the conclusion is the homeomorphy of the ~ech--Stone compactifications fiX1 and flX~ and of the Hewitt realcompactifications oX~ and oX2, respectively (cf. [5]): A. Let XI and X2 be Tikhonov spaces. (a) I f the semigroups C*(Xa) and C*(X~) are isomorphic then PX1 and BX~ are homeomorphic ; (b) if C(XO and C(X~) are isomorphic then oX1 and oX2 are homeomorphie.

Let X be a completely regular and Hausdorff topological space, v X its real compactification and Cc(X) [resp. C P (X)] the space of all continuous functions on X equipped with the topology of compact convergence [resp. of uniform convergence on a convenient family of subsets of X]. We give here the generalization to the case of the space C P (X), of the characterization of the barrelled, d-barrelled and σ-barrelled Cc(X) spaces.KeywordsContinuous FunctionComplex NumberCompact SubsetTopological Vector SpaceExtension TheoremThese 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 shown that any semilattice of bicompact G-extensions can be realized as a semilattice for a pseudocompact phase space with an action of a discrete group. The following examples are given: lattices of bicompact G-extensions which cannot be lattices of bicompact extensions of any Tikhonov space and a pseudocompact G-space with a discrete acting group whose semilattice of bicompact G-extensions has a countable number of minimal elements.

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.

A topological space is called a uqu space [10] if it admits a unique quasi-uniformity. Answering a question [2, Problem B, p. 45] of P. Fletcher and W. F. Lindgren in the affirmative we show in [8] that a topological space X is a uqu space if and only if every interior-preserving open collection of X is finite. (Recall that a collection ℒ of open sets of a topological space is called interior-preserving if the intersection of an arbitrary subcollection of ℒ is open (see e.g. [2, p. 29]).) The main step in the proof of this result in [8] shows that a topological space in which each interior-preserving open collection is finite is a transitive space. (A topological space is called transitive (see e.g. [2, p. 130]) if its fine quasi-uniformity has a base consisting of transitive entourages.) In the first section of this note we prove that each hereditarily compact space is transitive. The result of [8] mentioned above is an immediate consequence of this fact, because, obviously, a topological space in which each interior-preserving open collection is finite is hereditarily compact; see e.g. [2, Theorem 2.36]. Our method of proof also shows that a space is transitive if its fine quasi-uniformity is quasi-pseudo-metrizable. We use this result to prove that the fine quasi-uniformity of a T1 space X is quasi-metrizable if and only if X is a quasi-metrizable space containing only finitely many nonisolated points. This result should be compared with Proposition 2.34 of [2], which says that the fine quasi-uniformity of a regular T1 space has a countable base if and only if it is a metrizable space with only finitely many nonisolated points (see e.g. [11] for related results on uniformities). Another by-product of our investigations is the result that each topological space with a countable network is transitive.

Long colimits of topological groups II: Free groups and vector spaces

In many papers, new classes of sets had been studied in topological space, then the notion of continuity between any two topological spaces (a function from X to Y is continuous if the inverse image of each open set of Y is open in X) is studied via this new classes of sets. Here the authors also introduce new classes of sets called pj-b-preopen, pj-b-B set, pj-b-t set, pj-b-semi-open and pj-sb-generalized closed set in bitopological space [1] which is a set with two topologies defined on it, then they study the notion of continuity via this set and introduce some of the theories which are studying the decomposition of continuity via this set in bitopological space.

In this paper, we introduce a new concept called semi-perfect sets in topological spaces. We define semi-isolated points and semi-perfect sets. Further, some interesting results and properties of these concepts are investigated.

This paper introduces the notion of generalized hyperconnected sets in topological spaces using Levine’s concept of generalized closed sets. We define generalized hyperconnected sets and generalized hyperconnected components and study their fundamental properties. Furthermore, we provide necessary conditions for a topological space to be generalized hyperconnected. The relationship between this type of connectedness and other types is also presented. Several results concerning these new concepts are obtained and proven.

A fixed‐point theorem is proved under noncompact setting of general topological spaces. By applying the fixed‐point theorem, several new existence theorems of solutions for equilibrium problems are proved under noncompact setting of topological spaces. These theorems improve and generalize the corresponding results in related literature.

The concept main of paper is to connected between two definitions pre-open sets and g-open sets in the topological spaces is called Hpre-open sets, study restudy relations them, also present a new definitions, namely Hpre-continous, Hpre-open functions, Hpre-closed functions and have proved some theorems about these concept. Finally, we have present the definitions and results of separation axioms by Hpre-open sets,namely Hpre - T1 spaces , Hpre - T2 spaces and studied some of its applications.

Existence of solutions for quasi-equilibrium problems in noncompact topological spaces