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1
Content available remote

Commutativeness of Fundamental Groups of Topological Groups

100%
Korniłowicz A.
Formalized Mathematics
|
2013
|
tom 21
|
nr 2
127-131
EN
In this article we prove that fundamental groups based at the unit point of topological groups are commutative [11].
2
Content available remote

Examples of sequential topological groups under the continuum hypothesis

80%
Shibakov A.
Fundamenta Mathematicae
|
1996
|
tom 151
|
nr 2
107-120
EN
Using CH we construct examples of sequential topological groups: 1. a pair of countable Fréchet topological groups whose product is sequential but is not Fréchet, 2. a countable Fréchet and $α_1$ topological group which contains no copy of the rationals.
3
Content available remote

Topological groups and convex sets homeomorphic to non-separable Hilbert spaces

80%
Banakh T., Zarichnyy I.
Open Mathematics
|
2008
|
tom 6
|
nr 1
77-86
EN
Let X be a topological group or a convex set in a linear metric space. We prove that X is homeomorphic to (a manifold modeled on) an infinite-dimensional Hilbert space if and only if X is a completely metrizable absolute (neighborhood) retract with ω-LFAP, the countable locally finite approximation property. The latter means that for any open cover $$ \mathcal{U} $$ of X there is a sequence of maps (f n: X → X)nεgw such that each f n is $$ \mathcal{U} $$-near to the identity map of X and the family {f n(X)}n∈ω is locally finite in X. Also we show that a metrizable space X of density dens(X) < $$ \mathfrak{d} $$ is a Hilbert manifold if X has gw-LFAP and each closed subset A ⊂ X of density dens(A) < dens(X) is a Z ∞-set in X.
4
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Sequential topological groups of any sequential order under CH

80%
Shibakov A.
Fundamenta Mathematicae
|
1998
|
tom 155
|
nr 1
79-89
EN
For any $α < ω_1$ a countable sequential topological group of sequential order α is constructed using CH.
5
Content available remote

Topological Rough Groups

70%
Bağırmaz N., İçen İ., Özcan A. F.
Topological Algebra and its Applications
|
2016
|
tom 4
|
nr 1
EN
The concept of topological group is a simple combination of the concepts of abstract group and topological space. The purpose of this paper is to combine the concepts of topological space and rough groups; called topological rough groups on an approximation space.
6
Content available remote

Loop spaces and homotopy operations

61%
Blanc D.
Fundamenta Mathematicae
|
1997
|
tom 154
|
nr 1
75-95
EN
We describe an obstruction theory for an H-space X to be a loop space, in terms of higher homotopy operations taking values in $π_*X$. These depend on first algebraically "delooping" the Π-algebras $π_*X$, using the H-space structure on X, and then trying to realize the delooped Π-algebra.
7
Content available remote

On The Continuous Dependence Of Solutions To Orthogonal Additivity Problem On Given Functions

61%
Baron K.
Annales Mathematicae Silesianae
|
2015
|
tom 29
|
nr 1
19-23
EN
We show that the solution to the orthogonal additivity problem in real inner product spaces depends continuously on the given function and provide an application of this fact.
8
Content available remote

Uniform Space

61%
Coghetto R.
Formalized Mathematics
|
2016
|
tom 24
|
nr 3
215-226
EN
In this article, we formalize in Mizar [1] the notion of uniform space introduced by André Weil using the concepts of entourages [2]. We present some results between uniform space and pseudo metric space. We introduce the concepts of left-uniformity and right-uniformity of a topological group. Next, we define the concept of the partition topology. Following the Vlach’s works [11, 10], we define the semi-uniform space induced by a tolerance and the uniform space induced by an equivalence relation. Finally, using mostly Gehrke, Grigorieff and Pin [4] works, a Pervin uniform space defined from the sets of the form ((X\A) × (X\A)) ∪ (A×A) is presented.
9
Content available remote

Almost all submaximal groups are paracompact and σ-discrete

61%
Alas O. T., Protasov I. V., Tkačenko M. G., Tkachuk V. V., Wilson R. G., Yaschenko I. V.
Fundamenta Mathematicae
|
1998
|
tom 156
|
nr 3
241-260
EN
We prove that any topological group of a non-measurable cardinality is hereditarily paracompact and strongly σ-discrete as soon as it is submaximal. Consequently, such a group is zero-dimensional. Examples of uncountable maximal separable spaces are constructed in ZFC.
10
Content available remote

Summable Family in a Commutative Group

61%
Coghetto R.
Formalized Mathematics
|
2015
|
tom 23
|
nr 4
279-288
EN
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.
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