On orbits without the Baire property

2019 ◽  
Vol 26 (4) ◽  
pp. 625-628
Author(s):  
Alexander Kharazishvili
Keyword(s):  
Topological Space ◽  
Commutative Group ◽  
Baire Property ◽  
Group Of Homeomorphisms ◽  
Second Category

Abstract The following question is considered: when an uncountable commutative group of homeomorphisms of a second category topological space contains a subgroup, no orbit of which possesses the Baire property?

On the Complete Regularity of Some Category Spaces

1975 ◽  
Vol 17 (5) ◽  
pp. 651-656 ◽  
Author(s):  
W. Eames
Keyword(s):  
Topological Space ◽  
Density Function ◽  
Measure Space ◽  
Baire Property ◽  
Complete Regularity ◽  
Completely Regular ◽  
Upper Density ◽  
Covering Class ◽  
Sequential Covering ◽  
Null Set

A category space is a measure space which is also a topological space, the measure and the topology being related by ‘a set is measurable iff it has the Baire property’ and ‘a set is null iff it is nowhere dense’ [4]. We considered some category spaces in [3]; now we show that if a null set is deleted from the space, then the topology can be taken to be completely regular. The essential part of the construction consists of obtaining a suitable refinement of the original sequential covering class and using the consequent strong upper density function to define the required topology. Then the complete regularity follows much as in [1].

scholarly journals Summable Family in a Commutative Group

2015 ◽  
Vol 23 (4) ◽  
pp. 279-288
Author(s):  
Roland Coghetto
Keyword(s):  
Topological Space ◽  
Linear Space ◽  
Metric Space ◽  
Normed Linear Space ◽  
Linear Topological Space ◽  
Commutative Group ◽  
Image Filter ◽  
Directed Set ◽  
Generic Theory ◽  
Real Normed Linear Space

Summary Hölzl et al. showed that it was possible to build “a generic theory of limits based on filters” in Isabelle/HOL [22], [7]. In this paper we present our formalization of this theory in Mizar [6]. First, we compare the notions of the limit of a family indexed by a directed set, or a sequence, in a metric space [30], a real normed linear space [29] and a linear topological space [14] with the concept of the limit of an image filter [16]. Then, following Bourbaki [9], [10] (TG.III, §5.1 Familles sommables dans un groupe commutatif), we conclude by defining the summable families in a commutative group (“additive notation” in [17]), using the notion of filters.

scholarly journals On the regularity of topologies in the family of sets having the Baire property

Filomat ◽  
2013 ◽  
Vol 27 (7) ◽  
pp. 1291-1295 ◽  
Author(s):  
Jacek Hejduk
Keyword(s):  
Topological Space ◽  
Regularity Property ◽  
Baire Property ◽  
Proper Ideal ◽  
The Family ◽  
Meager Sets

The paper concerns the topologies introduced in the family of sets having the Baire property in a topological space (X, ?) and in the family generated by the sets having the Baire property and given a proper ?-ideal containing ? -meager sets. The regularity property of such topologies is investigated.

On topologies generated by some operators

2013 ◽  
Vol 11 (2) ◽  
Author(s):  
Katarzyna Flak ◽  
Jacek Hejduk
Keyword(s):  
Topological Space ◽  
Baire Property ◽  
Density Operators ◽  
The Family ◽  
Meager Sets

AbstractThe paper concerns topologies introduced in a topological space (X, τ) by operators which are much weaker than the lower density operators. Some properties of the family of sets having the Baire property and the family of meager sets with respect to such topologies are investigated.

scholarly journals σ-Algebra and σ-Baire in Fuzzy Soft Setting

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Shuker Mahmood Khalil ◽  
Mayadah Ulrazaq ◽  
Samaher Abdul-Ghani ◽  
Abu Firas Al-Musawi
Keyword(s):  
Topological Space ◽  
Basic Properties ◽  
Second Category ◽  
Soft Topological Space

We first introduce some new notions of Baireness in fuzzy soft topological space (FSTS). Next, their characterizations and basic properties are investigated in this work. The notions of fuzzy soft dense, fuzzy soft nowhere dense, fuzzy soft meager, fuzzy soft second category, fuzzy soft residual, fuzzy soft Baire, fuzzy soft δ-sets, fuzzy soft λσ-sets, fuzzy soft σ-nowhere dense, fuzzy soft σ-meager, fuzzy soft σ-residual, fuzzy soft σ-Baire, fuzzy soft σ-second category, fuzzy soft σ-residual, fuzzy, fuzzy soft submaximal space, fuzzy soft P-space, fuzzy soft almost resolvable space, fuzzy soft hyperconnected space, fuzzy soft A-embedded, fuzzy soft D-Baire, fuzzy soft almost P-spaces, fuzzy soft Borel, and fuzzy soft σ-algebra are introduced. Furthermore, several examples are shown as well.

scholarly journals Permutation representation of groups with Boolean orthogonalities

1981 ◽  
Vol 30 (4) ◽  
pp. 412-418
Author(s):  
Gary Davis
Keyword(s):  
Topological Space ◽  
Finiteness Condition ◽  
Permutation Representation ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Group Of Homeomorphisms

AbstractReduced rings and lattice-ordered groups are examples of groups with Boolean orthogonalities. In this note we show that any group with a Boolean orthogonality satisfying a finiteness condition introduced by Stewart is isomorphic with a group of homeomorphisms of a topological space, in which two homeomorphisms are orthogonal if and only if they have disjoint supports.

scholarly journals SOLUTIONS OF A GOŁA̧B–SCHINZEL-TYPE FUNCTIONAL EQUATION BOUNDED ON ‘BIG’ SETS IN AN ABSTRACT SENSE

2010 ◽  
Vol 81 (3) ◽  
pp. 430-441 ◽  
Author(s):  
ELIZA JABŁOŃSKA
Keyword(s):  
Functional Equation ◽  
Lebesgue Measure ◽  
Real Function ◽  
Baire Property ◽  
Positive Lebesgue Measure ◽  
Exponential Equation ◽  
Abstract Case ◽  
Second Category

AbstractIt is well known that an exponential real function, which is Lebesgue measurable (Baire measurable, respectively) or bounded on a set of positive Lebesgue measure (of the second category with the Baire property, respectively), is continuous. Here we consider bounded on ‘big’ set solutions of an equation generalizing the exponential equation as well as the Goła̧b–Schinzel equation. Moreover, we unify results into a more general and abstract case.

scholarly journals The boundedness principle characterizes second category subsets

1977 ◽  
Vol 16 (2) ◽  
pp. 257-265
Author(s):  
Kevin A. Broughan
Keyword(s):  
Topological Space ◽  
Lower Semicontinuous ◽  
Uniform Boundedness ◽  
Step Function ◽  
Lower Semicontinuous Function ◽  
Semicontinuous Function ◽  
Step Functions ◽  
Pointwise Limit ◽  
Second Category

Converses are proved for the Osgood (the Principle of Uniform Boundedness), Dini, and other well known. theorems. The notion of a continuous step function on a topological space is defined and a class of spaces identified for which each lower semicontinuous function is the pointwise limit of a monotonically increasing sequence of step functions.

scholarly journals A topological zero-one law for open continuous maps

1988 ◽  
Vol 38 (2) ◽  
pp. 263-266
Author(s):  
E. Barone ◽  
K.P.S. Bhaskara Rao
Keyword(s):  
Topological Space ◽  
Baire Property ◽  
Continuous Maps

We obtain a topological zero-one law for sets with the Baire property which are invariant under a semigroup of open continuous maps acting on a topological space.

Post-quantum digital signature schemes: setting a hidden group with two-dimensional cyclicity

2020 ◽  
Vol 4 ◽  
pp. 75-82
Author(s):  
D.Yu. Guryanov ◽  
◽  
D.N. Moldovyan ◽  
A. A. Moldovyan ◽  
Keyword(s):  
Associative Algebra ◽  
Digital Signature ◽  
Quantum Signature ◽  
Commutative Group ◽  
Two Dimensional ◽  
Signature Scheme ◽  
Signature Schemes ◽  
Fixed Element ◽  
Commutative Associative Algebra ◽  
Quantum Digital Signature

For the construction of post-quantum digital signature schemes that satisfy the strengthened criterion of resistance to quantum attacks, an algebraic carrier is proposed that allows one to define a hidden commutative group with two-dimensional cyclicity. Formulas are obtained that describe the set of elements that are permutable with a given fixed element. A post-quantum signature scheme based on the considered finite non-commutative associative algebra is described.

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