scholarly journals Antisymmetry of the Stochastical Order on all Ordered Topological Spaces

2019 ◽  
Vol 7 (1) ◽  
pp. 250-252 ◽  
Author(s):  
Tobias Fritz
Keyword(s):  
Topological Space ◽  
General Result ◽  
Elementary Proof ◽  
Short Note ◽  
Stochastic Order ◽  
Topological Spaces ◽  
Probability Measures ◽  
Ordered Topological Space ◽  
Special Cases

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.

Compactifications of ordered topological spaces

1972 ◽  
Vol 71 (3) ◽  
pp. 463-473 ◽  
Author(s):  
T. McCallion
Keyword(s):  
Topological Space ◽  
Partial Order ◽  
Topological Spaces ◽  
Ordered Topological Space

An ordered topological space is a set X endowed with a topology τ and a partial order ≤. We shall denote such a space by (X, τ), it being understood that (unless otherwise stated) the symbol ≤ is used to denote the partial order on X. An account of some of the properties of these spaces can be found in (5), (6) and (7).

Compactifications of discrete versions of semitopological semigroups by filters of zero sets

1991 ◽  
Vol 109 (2) ◽  
pp. 363-373
Author(s):  
Talin Budak (Papazyan)
Keyword(s):  
General Theory ◽  
Topological Space ◽  
Topological Spaces ◽  
Dense Subspace ◽  
Hausdorff Topology ◽  
Zero Sets ◽  
Compact Topological Space ◽  
Function Algebras ◽  
Special Cases ◽  
Discrete Semigroups

AbstractThe maximal proper prime filters together with the ultrafilters of zero sets of any metrizable compact topological space are shown to have a compact Hausdorff topology in which the ultrafilters form a discrete, dense subspace. This gives a general theory of compactifications of discrete versions of compact metrizable topological spaces and some of the already known constructions of compact right topological semigroups are special cases of the general theory. In this way, simpler and more elegant proofs for these constructions are obtained.In [8], Pym constructed compactifications for discrete semigroups which can be densely embedded in a compact group. His techniques made extensive use of function algebras. In [4] Helmer and Isik obtained the same compactifications by using the existence of Stone ech compactifications. The aim of this paper is to present a general theory of compactifications of semitopological semigroups so that Helmer and Isik's results in [4] are a simple consequence. Our proofs are different and are based on filters which provide a natural way of getting compactifications. Moreover we present new insights by emphasizing maximal proper primes which are not ultrafilters.We start by defining filters of zero sets (called z-filters) on a given topological space X, and their convergence. In the case of compact metrizable topological spaces, we establish the connections between proper maximal prime z-filters on X and zultrafilters in β(X\{x})\(X\{x}) where β(X\{x}) is the Stone-ech compactification of X\{x}. We then define a topology on the set of all prime z-filters on X such that the subspace of all proper maximal primes is compact Hausdorff. We denote by the set of all proper maximal prime z-filters on X together with the z-ultrafilters and show that when X is a compact metrizable cancellative semitopological semigroup, is a compact right topological semigroup with dense topological centre. Also, when is considered for a compact Hausdorff metrizable group, the semigroup obtained is exactly the same (algebraically and topologically) as the semigroup obtained in [4]. Hence the result in [4] is just a consequence of the general theory presented in this paper.

On topological quotient maps preserved by pullbacks or products

1970 ◽  
Vol 67 (3) ◽  
pp. 553-558 ◽  
Author(s):  
B. J. Day ◽  
G. M. Kelly
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Quotient Maps ◽  
Continuous Maps

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?

scholarly journals Separation Axioms in Intuitionistic Fuzzy Topological Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-7
Author(s):  
Amit Kumar Singh ◽  
Rekha Srivastava
Keyword(s):  
Topological Space ◽  
Left Adjoint ◽  
Topological Spaces ◽  
Intuitionistic Fuzzy ◽  
Separation Axioms ◽  
Fuzzy Topological Space ◽  
Fuzzy Topological Spaces

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.

Optimal allocation of relevations in coherent systems

2021 ◽  
Vol 58 (4) ◽  
pp. 1152-1169
Author(s):  
Rongfang Yan ◽  
Jiandong Zhang ◽  
Yiying Zhang
Keyword(s):  
Optimal Allocation ◽  
Necessary Conditions ◽  
Stochastic Order ◽  
Coherent Systems ◽  
Stochastic Comparisons ◽  
Hazard Rate Order ◽  
Usual Stochastic Order ◽  
Special Cases ◽  
Sufficient And Necessary Conditions ◽  
Allocation Strategies

AbstractIn this paper we study the allocation problem of relevations in coherent systems. The optimal allocation strategies are obtained by implementing stochastic comparisons of different policies according to the usual stochastic order and the hazard rate order. As special cases of relevations, the load-sharing and minimal repair policies are further investigated. Sufficient (and necessary) conditions are established for various stochastic orderings. Numerical examples are also presented as illustrations.

Uniformity on generalized topological spaces

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Dipankar Dey ◽  
Dhananjay Mandal ◽  
Manabendra Nath Mukherjee
Keyword(s):  
Topological Space ◽  
Present Article ◽  
Design Methodology ◽  
Original Work ◽  
Generalized Topology ◽  
Topological Spaces ◽  
Intermediate Structure ◽  
Content Type ◽  
Completely Regular ◽  
Generalized Topological Space

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.

scholarly journals Study of a Forwarding Chain in the Category of Topological Spaces between T0 and T2 with respect to One Point Compactification Operator

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Fatemah Ayatollah Zadeh Shirazi ◽  
Meysam Miralaei ◽  
Fariba Zeinal Zadeh Farhadi
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
The One ◽  
One Point Compactification

In the following text, we want to study the behavior of one point compactification operator in the chain Ξ := {Metrizable, Normal, T2, KC, SC, US, T1, TD, TUD, T0, Top} of subcategories of category of topological spaces, Top (where we denote the subcategory of Top, containing all topological spaces with property P , simply by P). Actually we want to know, for P∈Ξ and X∈P, the one point compactification of topological space X belongs to which elements of Ξ. Finally we find out that the chain {Metrizable, T2, KC, SC, US, T1, TD, TUD, T0, Top} is a forwarding chain with respect to one point compactification operator.

scholarly journals Orderability of topological spaces by continuous preferences

2001 ◽  
Vol 27 (8) ◽  
pp. 505-512 ◽  
Author(s):  
José Carlos Rodríguez Alcantud
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Topological Properties ◽  
Linear Orders ◽  
Mathematical Economics

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.

scholarly journals A Tychonoff theorem in intuitionistic fuzzy topological spaces

2004 ◽  
Vol 2004 (70) ◽  
pp. 3829-3837
Author(s):  
Doğan Çoker ◽  
A. Haydar Eş ◽  
Necla Turanli
Keyword(s):  
Topological Space ◽  
Fuzzy Set ◽  
Intuitionistic Fuzzy Set ◽  
Topological Spaces ◽  
Fuzzy Topology ◽  
Intuitionistic Fuzzy ◽  
Fuzzy Topological Space ◽  
Fuzzy Topological Spaces ◽  
Tychonoff Theorem

The purpose of this paper is to prove a Tychonoff theorem in the so-called “intuitionistic fuzzy topological spaces.” After giving the fundamental definitions, such as the definitions of intuitionistic fuzzy set, intuitionistic fuzzy topology, intuitionistic fuzzy topological space, fuzzy continuity, fuzzy compactness, and fuzzy dicompactness, we obtain several preservation properties and some characterizations concerning fuzzy compactness. Lastly we give a Tychonoff-like theorem.

scholarly journals A glance into the anatomy of monotonic maps

2021 ◽  
Vol 22 (1) ◽  
pp. 1
Author(s):  
Raushan Buzyakova
Keyword(s):  
Topological Space ◽  
Ordered Topological Space ◽  
Original Space ◽  
Ordered Space

<p>Given an autohomeomorphism on an ordered topological space or its subspace, we show that it is sometimes possible to introduce a new topology-compatible order on that space so that the same map is monotonic with respect to the new ordering. We note that the existence of such a re-ordering for a given map is equivalent to the map being conjugate (topologically equivalent) to a monotonic map on some homeomorphic ordered space. We observe that the latter cannot always be chosen to be order-isomorphic to the original space. Also, we identify other routes that may lead to similar affirmative statements for other classes of spaces and maps.</p>

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