On groups of essential values of topological cylinder cocycles over minimal rotations

Mieczysław K. Mentzen

Displaying similar documents to “On groups of essential values of topological cylinder cocycles over minimal rotations”

Arhangel'skiĭ sheaf amalgamations in topological groups

Boaz Tsaban, Lyubomyr Zdomskyy (2016)

Fundamenta Mathematicae

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We consider amalgamation properties of convergent sequences in topological groups and topological vector spaces. The main result of this paper is that, for arbitrary topological groups, Nyikos’s property $α_{1.5}$ is equivalent to Arhangel’skiĭ’s formally stronger property α₁. This result solves a problem of Shakhmatov (2002), and its proof uses a new perturbation argument. We also prove that there is a topological space X such that the space $C_{p} (X)$ of continuous real-valued functions on X with the...

Cellularity and the index of narrowness in topological groups

Mihail G. Tkachenko (2011)

Commentationes Mathematicae Universitatis Carolinae

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We study relations between the cellularity and index of narrowness in topological groups and their $G_{δ}$ -modifications. We show, in particular, that the inequalities $in ({(H)}_{τ}) \leq 2^{τ \cdot in (H)}$ and $c ({(H)}_{τ}) \leq 2^{2^{τ \cdot in (H)}}$ hold for every topological group $H$ and every cardinal $τ \geq ω$ , where ${(H)}_{τ}$ denotes the underlying group $H$ endowed with the $G_{τ}$ -modification of the original topology of $H$ and $in (H)$ is the index of narrowness of the group $H$ . Also, we find some bounds for the complexity of continuous real-valued functions $f$ on an arbitrary $ω$ -narrow group...

The universal minimal system for the group of homeomorphisms of the Cantor set

E. Glasner, B. Weiss (2003)

Fundamenta Mathematicae

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Each topological group G admits a unique universal minimal dynamical system (M(G),G). For a locally compact noncompact group this is a nonmetrizable system with a rich structure, on which G acts effectively. However there are topological groups for which M(G) is the trivial one-point system (extremely amenable groups), as well as topological groups G for which M(G) is a metrizable space and for which one has an explicit description. We show that for the topological group G = Homeo(E)...

On topological groups with a small base and metrizability

Saak Gabriyelyan, Jerzy Kąkol, Arkady Leiderman (2015)

Fundamenta Mathematicae

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A (Hausdorff) topological group is said to have a -base if it admits a base of neighbourhoods of the unit, $U_{α} : α \in ℕ^{ℕ}$ , such that $U_{α} \subset U_{β}$ whenever β ≤ α for all $α, β \in ℕ^{ℕ}$ . The class of all metrizable topological groups is a proper subclass of the class $T G_{}$ of all topological groups having a -base. We prove that a topological group is metrizable iff it is Fréchet-Urysohn and has a -base. We also show that any precompact set in a topological group $G \in T G_{}$ is metrizable, and hence G is strictly angelic. We deduce from...

Ordinals in topological groups

Raushan Z. Buzyakova (2007)

Fundamenta Mathematicae

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We show that if an uncountable regular cardinal τ and τ + 1 embed in a topological group G as closed subspaces then G is not normal. We also prove that an uncountable regular cardinal cannot be embedded in a torsion free Abelian group that is hereditarily normal. These results are corollaries to our main results about ordinals in topological groups. To state the main results, let τ be an uncountable regular cardinal and G a T₁ topological group. We prove, among others, the following...

Ultra-b-barrelled spaces and the completeness of $L_{b} (E, F)$

Z. Kadelburg (1979)

Matematički Vesnik

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Extremal phenomena in certain classes of totally bounded groups

W. W. Comfort, Lewis C. Robertson

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For various pairs of topological properties such that P ⇒ Q, we consider two questions: (A) Does every topological group topology with P extend properly to a topological group topology with Q, and (B) must a topological group with P have a proper dense subgroup with Q? We obtain negative results and positive results. Principal among the latter is the statement that any pseudocompact group G of uncountable weight which satisfies any of the following three conditions has both a strictly...

A note on $R_{0}$ -topological spaces

K. K. Dube (1974)

Matematički Vesnik

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Nonnormality of remainders of some topological groups

Aleksander V. Arhangel&#039;skii, J. van Mill (2016)

Commentationes Mathematicae Universitatis Carolinae

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It is known that every remainder of a topological group is Lindelöf or pseudocompact. Motivated by this result, we study in this paper when a topological group $G$ has a normal remainder. In a previous paper we showed that under mild conditions on $G$ , the Continuum Hypothesis implies that if the Čech-Stone remainder $G^{*}$ of $G$ is normal, then it is Lindelöf. Here we continue this line of investigation, mainly for the case of precompact groups. We show that no pseudocompact group, whose weight...

On sequences over a finite abelian group with zero-sum subsequences of forbidden lengths

Weidong Gao, Yuanlin Li, Pingping Zhao, Jujuan Zhuang (2016)

Colloquium Mathematicae

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Let G be an additive finite abelian group. For every positive integer ℓ, let $d i s c_{ℓ} (G)$ be the smallest positive integer t such that each sequence S over G of length |S| ≥ t has a nonempty zero-sum subsequence of length not equal to ℓ. In this paper, we determine $d i s c_{ℓ} (G)$ for certain finite groups, including cyclic groups, the groups $G = C ₂ \oplus C_{2 m}$ and elementary abelian 2-groups. Following Girard, we define disc(G) as the smallest positive integer t such that every sequence S over G with |S| ≥ t has nonempty zero-sum...

Cylinder cocycle extensions of minimal rotations on monothetic groups

Mieczysław K. Mentzen, Artur Siemaszko (2004)

Colloquium Mathematicae

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The main results of this paper are: 1. No topologically transitive cocycle $ℝ^{m}$ -extension of minimal rotation on the unit circle by a continuous real-valued bounded variation ℤ-cocycle admits minimal subsets. 2. A minimal rotation on a compact metric monothetic group does not admit a topologically transitive real-valued cocycle if and only if the group is finite.

Minimal systems and distributionally scrambled sets

Piotr Oprocha (2012)

Bulletin de la Société Mathématique de France

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In this paper we investigate numerous constructions of minimal systems from the point of view of $(ℱ_{1}, ℱ_{2})$ -chaos (but most of our results concern the particular cases of distributional chaos of type $1$ and $2$ ). We consider standard classes of systems, such as Toeplitz flows, Grillenberger $K$ -systems or Blanchard-Kwiatkowski extensions of the Chacón flow, proving that all of them are DC2. An example of DC1 minimal system with positive topological entropy is also introduced. The above mentioned results...

Algebra in the superextensions of twinic groups

Taras Banakh, Volodymyr Gavrylkiv

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Given a group X we study the algebraic structure of the compact right-topological semigroup λ(X) consisting of all maximal linked systems on X. This semigroup contains the semigroup β(X) of ultrafilters as a closed subsemigroup. We construct a faithful representation of the semigroup λ(X) in the semigroup ${(X)}^{(X)}$ of all self-maps of the power-set (X) and show that the image of λ(X) in ${(X)}^{(X)}$ coincides with the semigroup $E n d_{λ} ((X))$ of all functions f: (X) → (X) that are equivariant, monotone and symmetric...

A note on minimal zero-sum sequences over ℤ

Papa A. Sissokho (2014)

Acta Arithmetica

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A zero-sum sequence over ℤ is a sequence with terms in ℤ that sum to 0. It is called minimal if it does not contain a proper zero-sum subsequence. Consider a minimal zero-sum sequence over ℤ with positive terms $a ₁, . . ., a_{h}$ and negative terms $b ₁, . . ., b_{k}$ . We prove that h ≤ ⌊σ⁺/k⌋ and k ≤ ⌊σ⁺/h⌋, where $σ ⁺ = \sum_{i = 1}^{h} a_{i} = - \sum_{j = 1}^{k} b_{j}$ . These bounds are tight and improve upon previous results. We also show a natural partial order structure on the collection of all minimal zero-sum sequences over the set i∈ ℤ : -n ≤ i ≤ n for any positive...

Group Structures and Rectifiability in Powers of Spaces

G. J. Ridderbos (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

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We prove that if some power of a space X is rectifiable, then $X^{π w (X)}$ is rectifiable. It follows that no power of the Sorgenfrey line is a topological group and this answers a question of Arhangel’skiĭ. We also show that in Mal’tsev spaces of point-countable type, character and π-character coincide.

Linear topological properties of the Lumer-Smirnov class of the polydisc

Marek Nawrocki (1992)

Studia Mathematica

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Linear topological properties of the Lumer-Smirnov class $L N_{*} (^{n})$ of the unit polydisc $^{n}$ are studied. The topological dual and the Fréchet envelope are described. It is proved that $L N_{*} (^{n})$ has a weak basis but it is nonseparable in its original topology. Moreover, it is shown that the Orlicz-Pettis theorem fails for $L N_{*} (^{n})$ .

Remarks on flat and differential $K$ -theory

Man-Ho Ho (2014)

Annales mathématiques Blaise Pascal

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In this note we prove some results in flat and differential $K$ -theory. The first one is a proof of the compatibility of the differential topological index and the flat topological index by a direct computation. The second one is the explicit isomorphisms between Bunke-Schick differential $K$ -theory and Freed-Lott differential $K$ -theory.

On the Hausdorff Dimension of Topological Subspaces

Tomasz Szarek, Maciej Ślęczka (2006)

Bulletin of the Polish Academy of Sciences. Mathematics

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It is shown that every Polish space X with $d i m_{T} X \geq d$ admits a compact subspace Y such that $d i m_{H} Y \geq d$ where $d i m_{T}$ and $d i m_{H}$ denote the topological and Hausdorff dimensions, respectively.

On some properties of three-dimensional minimal sets in $ℝ^{4}$

Tien Duc Luu (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

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We prove in this paper the Hölder regularity of Almgren minimal sets of dimension 3 in $ℝ^{4}$ around a $𝕐$ -point and the existence of a point of particular type of a Mumford-Shah minimal set in $ℝ^{4}$ , which is very close to a $𝕋$ . This will give a local description of minimal sets of dimension 3 in $ℝ^{4}$ around a singular point and a property of Mumford-Shah minimal sets in $ℝ^{4}$ .

On a translation property of positive definite functions

Lars Omlor, Michael Leinert (2010)

Banach Center Publications

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If G is a locally compact group with a compact invariant neighbourhood of the identity e, the following property (*) holds: For every continuous positive definite function h≥ 0 with compact support there is a constant $C_{h} > 0$ such that $\int L_{x} h \cdot g \leq C_{h} \int h g$ for every continuous positive definite g≥0, where $L_{x}$ is left translation by x. In [L], property (*) was stated, but the above inequality was proved for special h only. That “for one h” implies “for all h” seemed obvious, but turned out not to be obvious at...