scholarly journals Some types of fuzzy open sets in fuzzy topological groups

2015 ◽  
Vol 3 (2) ◽  
pp. 103
Author(s):  
Mohamad Thigeel Hmod ◽  
Munir A. Al-khafaji ◽  
Taghreed Hur Majeed
Keyword(s):  
Topological Groups ◽  
Compact Spaces ◽  
Basic Concepts ◽  
Regular Open

<p>The aim of this work is to introduce the definitions and study the concepts of fuzzy open (resp, fuzzy α- open, fuzzy semi- open, fuzzy pre- open, fuzzy regular- open, fuzzy b- open, fuzzy β- open) sets in fuzzy topological groups, and devote to study and discuss some of the basic concepts of some types of fuzzy continuous, fuzzy connected and fuzzy compact spaces in fuzzy topological groups with some theorems and Proposition are proved.</p>

On the Dimension Modulo Classes of Topological Spaces and Free Topological Groups

2003 ◽  
Vol 10 (2) ◽  
pp. 209-222
Author(s):  
I. Bakhia
Keyword(s):  
Wide Class ◽  
Sufficient Conditions ◽  
Topological Spaces ◽  
Topological Groups ◽  
Compact Spaces ◽  
Topological Semigroups

Abstract Functions of dimension modulo a (rather wide) class of spaces are considered and the conditions are found, under which the dimension of the product of spaces modulo these classes is equal to zero. Based on these results, the sufficient conditions are established, under which spaces of free topological semigroups (in the sense of Marxen) and spaces of free topological groups (in the sense of Markov and Graev) are zero-dimensional modulo classes of compact spaces.

scholarly journals The completion of a topological group

1980 ◽  
Vol 21 (3) ◽  
pp. 407-417 ◽  
Author(s):  
Eric C. Nummela
Keyword(s):  
Topological Group ◽  
Topological Spaces ◽  
Topological Groups ◽  
Russian School ◽  
Compact Spaces ◽  
French School ◽  
Absolutely Closed

During the 1920's and 30's, two distinct theories of “completions” for topological spaces were being developed: the French school of mathematics was describing the familiar notion of “complete relative to a uniformity”, and the Russian school the less well-known idea of “absolutely closed”. The two agree precisely for compact spaces.The first part of this article describes these two notions of completeness; the remainder is a presentation of the interesting, but apparently unrecorded, fact that the two ideas coincide when put in the context of topological groups.

A Note on Small Baire Spaces

1986 ◽  
Vol 38 (3) ◽  
pp. 659-665 ◽  
Author(s):  
Saharon Shelah ◽  
Stevo Todorcevic
Keyword(s):  
Topological Space ◽  
Metric Spaces ◽  
Baire Category ◽  
Baire Space ◽  
Natural Question ◽  
Baire Category Theorem ◽  
Complete Metric Spaces ◽  
Compact Spaces ◽  
Category Theorem ◽  
Regular Open

A Baire space is a topological space which satisfies the Baire Category Theorem, i.e., in which the intersection of countably many dense open sets is dense. In this note we shall be interested in the size of Baire spaces, so to avoid trivialities we shall consider only non-atomic spaces, that is, spaces X whose regular open algebras ro(X) are non-atomic. All natural examples of Baire spaces, such as complete metric spaces or compact spaces, seem to have sizes at least 2ℵ0. So a natural question, asked first by W. Fleissner and K. Kunen [5], is whether there exists a Baire space of the minimal possible size ℵ1. The purpose of this note is to show that such a space need not exist by proving the following result.

scholarly journals Separability of Topological Groups: A Survey with Open Problems

Axioms ◽  
2018 ◽  
Vol 8 (1) ◽  
pp. 3 ◽  
Author(s):  
Arkady Leiderman ◽  
Sidney Morris
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Quotient Space ◽  
Topological Groups ◽  
Open Problems ◽  
Infinite Products ◽  
Matrix Groups ◽  
Compact Spaces ◽  
Finite Dimensional ◽  
Separable Quotient Problem

Separability is one of the basic topological properties. Most classical topological groups and Banach spaces are separable; as examples we mention compact metric groups, matrix groups, connected (finite-dimensional) Lie groups; and the Banach spaces C ( K ) for metrizable compact spaces K; and ℓ p , for p ≥ 1 . This survey focuses on the wealth of results that have appeared in recent years about separable topological groups. In this paper, the property of separability of topological groups is examined in the context of taking subgroups, finite or infinite products, and quotient homomorphisms. The open problem of Banach and Mazur, known as the Separable Quotient Problem for Banach spaces, asks whether every Banach space has a quotient space which is a separable Banach space. This paper records substantial results on the analogous problem for topological groups. Twenty open problems are included in the survey.

Separable Quotients of Free Topological Groups

2019 ◽  
Vol 63 (3) ◽  
pp. 610-623 ◽  
Author(s):  
Arkady Leiderman ◽  
Mikhail Tkachenko
Keyword(s):  
Topological Group ◽  
Point Group ◽  
Topological Groups ◽  
Topological Abelian Group ◽  
Abelian Topological Group ◽  
Compact Spaces ◽  
Separable Subgroup ◽  
Pseudocompact Spaces ◽  
Free Abelian Topological Group ◽  
The One

AbstractWe study the following problem: For which Tychonoff spaces $X$ do the free topological group $F(X)$ and the free abelian topological group $A(X)$ admit a quotient homomorphism onto a separable and nontrivial (i.e., not finitely generated) group? The existence of the required quotient homomorphisms is established for several important classes of spaces $X$, which include the class of pseudocompact spaces, the class of locally compact spaces, the class of $\unicode[STIX]{x1D70E}$-compact spaces, the class of connected locally connected spaces, and some others.We also show that there exists an infinite separable precompact topological abelian group $G$ such that every quotient of $G$ is either the one-point group or contains a dense non-separable subgroup and, hence, does not have a countable network.

scholarly journals Unions of chains in dyadic compact spaces and topological groups

2002 ◽  
Vol 121 (1-2) ◽  
pp. 25-32 ◽  
Author(s):  
Mikhail G. Tkachenko ◽  
Yolanda Torres Falcón
Keyword(s):  
Topological Groups ◽  
Compact Spaces

scholarly journals Dense orbits and compactness of groups

2003 ◽  
Vol 68 (1) ◽  
pp. 155-159 ◽  
Author(s):  
W. H. Previts ◽  
T. S. Wu
Keyword(s):  
Haar Measure ◽  
Topological Dynamics ◽  
Topological Groups ◽  
Locally Compact ◽  
Group Automorphism ◽  
Compact Spaces ◽  
Dense Orbits ◽  
Locally Compact Spaces ◽  
Totally Disconnected

Using results from topological groups and topological dynamics for locally compact spaces, Aoki has shown that when a group automorphism of a locally compact totally disconnected group is ergodic under the Haar measure, the group is compact. We provide some remarks on Aoki's proof. Also we present a new proof of his result using the structure of locally compact totally disconnected groups established by Willis.

scholarly journals An operation on topological spaces

2000 ◽  
Vol 1 (1) ◽  
pp. 13
Author(s):  
A.V. Arhangelskii
Keyword(s):  
Topological Space ◽  
Topological Structure ◽  
Necessary Conditions ◽  
Topological Spaces ◽  
Topological Groups ◽  
Compact Spaces ◽  
Product Operation ◽  
Made In

<p>A (binary) product operation on a topological space X is considered. The only restrictions are that some element e of X is a left and a right identity with respect to this multiplication, and that certain natural continuity requirements are satisfied. The operation is called diagonalization (of X). Two problems are considered: 1. When a topological space X admits such an operation, that is, when X is diagonalizable? 2. What are necessary conditions for diagonalizablity of a space (at a given point)? A progress is made in the article on both questions. In particular, it is shown that certain deep results about the topological structure of compact topological groups can be extended to diagonalizable compact spaces. The notion of a Moscow space is instrumental in our study.</p>

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