scholarly journals A VIEW TO POWER SET: A TOPOLOGICAL DISCUSSION

2019 ◽  
pp. 1-2
Author(s):  
aripex Amuly

A Power Set is not only a container of all family of subsets of a set and the set itself,but ,in topology,it is also a generator of all topologies on the defined set. So, there is a topological existence of power set, being the strongest topology ever defined on a set,there are some properties of it's topological existence.In this paper, such properties are being proved and concluded. The following theorems stated are on the basis of the topological properties and separated axioms,which by satisfying, moves to a conclusion that,not only a power set is just a topology on the given defined set,but also it can be considered as a “Universal Topology”or a “Universal Topological Space”,that is the container of all topological spaces. This paper gives a general understanding about what a power set is,topologically and gives us a new perceptive from a “power set”to a “topological power space”.

2001 ◽  
Vol 27 (8) ◽  
pp. 505-512 ◽  
Author(s):  
José Carlos Rodríguez Alcantud

We extend van Dalen and Wattel's (1973) characterization of orderable spaces and their subspaces by obtaining analogous results for two larger classes of topological spaces. This type of spaces are defined by considering preferences instead of linear orders in the former definitions, and possess topological properties similar to those of (totally) orderable spaces (cf. Alcantud, 1999). Our study provides particular consequences of relevance in mathematical economics; in particular, a condition equivalent to the existence of a continuous preference on a topological space is obtained.


2016 ◽  
Vol 12 (4) ◽  
pp. 6178-6184 ◽  
Author(s):  
A A Nasef ◽  
A E Radwan ◽  
F A Ibrahem ◽  
R B Esmaeel

In the present paper, we have continued to study the properties of soft topological spaces. We introduce new types of soft compactness based on the soft ideal Ĩ in a soft topological space (X, τ, E) namely, soft αI-compactness, soft αI-Ĩ-compactness, soft α-Ĩ-compactness, soft α-closed, soft αI-closed, soft countably α-Ĩ-compactness and soft countably αI-Ĩ-compactness. Also, several of their topological properties are investigated. The behavior of these concepts under various types of soft functions has obtained


1981 ◽  
Vol 31 (4) ◽  
pp. 385-389
Author(s):  
Graham J. Logan

AbstractThis paper discusses two notions, developed independently and both termed “cocompactness”. The first arises in the area of topology, where J. de Groot and others have studied spaces which are, in a certain sense, complementary to a given space. If the given space is compact then the complementary spaces are said to be cocompact. The second concept arises in the area of logic and general algebra. Loosely speaking a logic is compact if every inconsistent set of formulas has a finite inconsistent subset. This notion of compactness may be generalized to any closure algebra and the use of the term “cocompactness” to describe the generalization was suggested to the author by Dr. R. A. Bull.It is shown here that topological and algebraic cocompactness are related in the following ways. Firstly, if a closure algebra is algebraically cocompact then its dual space is topologjcally cocompact, and conditions may be given for the implication to be reversible.3 Furthermore any cocompact topological space may be represented as the continuous 1-1 image of the dual space of a cocompact closure algebra. A final result relates another class of closure algebras with those topological spaces which are compact.


Author(s):  
Ali Kandil ◽  
Osama A. El-Tantawy ◽  
Sobhy A. El-Sheikh ◽  
A. M. Abd El-latif

The main purpose of this chapter is to introduce the notions of ?-operation, pre-open soft set a-open sets, semi open soft set and ß-open soft sets to soft topological spaces. We study the relations between these different types of subsets of soft topological spaces. We introduce new soft separation axioms based on the semi open soft sets which are more general than of the open soft sets. We show that the properties of soft semi Ti-spaces (i=1,2) are soft topological properties under the bijection and irresolute open soft mapping. Also, we introduce the notion of supra soft topological spaces. Moreover, we introduce the concept of supra generalized closed soft sets (supra g-closed soft for short) in a supra topological space (X,µ,E) and study their properties in detail.


Author(s):  
G. Bosi ◽  
A. Estevan ◽  
J. Gutiérrez García ◽  
E. Induráin

In this paper, we go further on the problem of the continuous numerical representability of interval orders defined on topological spaces. A new condition of compatibility between the given topology and the indifference associated to the main trace of an interval order is introduced. Provided that this condition is fulfilled, a semiorder has a continuous interval order representation through a pair of continuous real-valued functions. Other necessary and sufficient conditions for the continuous representability of interval orders are also discussed, and, in particular, a characterization is achieved for the particular case of interval orders defined on a topological space of finite support.


2021 ◽  
Vol 13(62) (2) ◽  
pp. 683-696
Author(s):  
Karishma Shravan ◽  
Binod Chandra Tripathy

In this paper, we have investigated one of the basic topological properties, called Metrizability in multiset topological space. Metrizable spaces are those topological spaces which are homeomorphic to a metric space. So, we first give the notion of metric between two multi-points in a finite multiset and studied some significant properties of a multiset metric space. The notion of metrizability is then studied by using this metric. Besides, the Urysohn’s lemma which is considered to be one of the important tools in studying some metrization theorems in topology is also discussed in context with multisets.


2019 ◽  
Vol 7 (1) ◽  
pp. 250-252 ◽  
Author(s):  
Tobias Fritz

Abstract In this short note, we prove that the stochastic order of Radon probability measures on any ordered topological space is antisymmetric. This has been known before in various special cases. We give a simple and elementary proof of the general result.


Author(s):  
B. J. Day ◽  
G. M. Kelly

We are concerned with the category of topological spaces and continuous maps. A surjection f: X → Y in this category is called a quotient map if G is open in Y whenever f−1G is open in X. Our purpose is to answer the following three questions:Question 1. For which continuous surjections f: X → Y is every pullback of f a quotient map?Question 2. For which continuous surjections f: X → Y is f × lz: X × Z → Y × Z a quotient map for every topological space Z? (These include all those f answering to Question 1, since f × lz is the pullback of f by the projection map Y ×Z → Y.)Question 3. For which topological spaces Z is f × 1Z: X × Z → Y × Z a qiptoent map for every quotient map f?


2012 ◽  
Vol 2012 ◽  
pp. 1-7
Author(s):  
Amit Kumar Singh ◽  
Rekha Srivastava

In this paper we have studied separation axiomsTi,i=0,1,2in an intuitionistic fuzzy topological space introduced by Coker. We also show the existence of functorsℬ:IF-Top→BF-Topand𝒟:BF-Top→IF-Topand observe that𝒟is left adjoint toℬ.


2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Dipankar Dey ◽  
Dhananjay Mandal ◽  
Manabendra Nath Mukherjee

PurposeThe present article deals with the initiation and study of a uniformity like notion, captioned μ-uniformity, in the context of a generalized topological space.Design/methodology/approachThe existence of uniformity for a completely regular topological space is well-known, and the interrelation of this structure with a proximity is also well-studied. Using this idea, a structure on generalized topological space has been developed, to establish the same type of compatibility in the corresponding frameworks.FindingsIt is proved, among other things, that a μ-uniformity on a non-empty set X always induces a generalized topology on X, which is μ-completely regular too. In the last theorem of the paper, the authors develop a relation between μ-proximity and μ-uniformity by showing that every μ-uniformity generates a μ-proximity, both giving the same generalized topology on the underlying set.Originality/valueIt is an original work influenced by the previous works that have been done on generalized topological spaces. A kind of generalization has been done in this article, that has produced an intermediate structure to the already known generalized topological spaces.


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