Topology and its Applications最新文献

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Quasi-uniform entropy vs topological entropy 准均匀熵与拓扑熵

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2024-08-28 DOI: 10.1016/j.topol.2024.109054

Paulus Haihambo , O. Olela Otafudu

In 2023 Haihambo and Olela Otafudu introduced and studied the notion of quasi-uniform entropy $h_{Q U} (ψ)$ for a uniformly continuous self-map ψ of a quasi-metric or a quasi-uniform space X. In this paper, we discuss the connection between the topological entropy functions $h, h_{f}$ and the quasi-uniform entropy function $h_{Q U}$ on a quasi-uniform space X, where h and $h_{f}$ are the topological entropy functions defined using compact sets and finite open covers, respectively. In particular, we have shown that for a uniformly continuous self-map ψ of a $T_{0}$ -quasi-uniform space $(X, U)$ we have $h (ψ) \leq h_{Q U} (ψ)$ when X is compact and $h_{Q U} (ψ) \leq h_{f} (ψ)$ with equality if X is a compact $T_{2}$ space.

2023 年，Haihambo 和 Olela Otafudu 提出并研究了准度量空间或准均匀空间 X 的均匀连续自映射 ψ 的准均匀熵 hQU(ψ) 概念。本文讨论了拓扑熵函数 h,hf 与准均匀空间 X 上的准均匀熵函数 hQU 之间的联系，其中 h 和 hf 分别是用紧凑集和有限开盖定义的拓扑熵函数。特别是，我们已经证明，对于 T0-准均匀空间 (X,U) 的均匀连续自映射 ψ，当 X 紧凑时，有 h(ψ)≤hQU(ψ) ；如果 X 是紧凑的 T2 空间，则 hQU(ψ)≤hf(ψ) 相等。

引用次数: 0

On aspherical configuration Lie groupoids 关于非球面构型李群

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2024-08-28 DOI: 10.1016/j.topol.2024.109052

S.K. Roushon

The complement of the hyperplanes ${x_{i} = x_{j}}$ , for all $i \neq j$ , in $M^{n}$ , where M is an aspherical 2-manifold, is known to be aspherical. Here we consider the situation when M is a 2-dimensional orbifold. We prove this complement to be aspherical for a class of aspherical 2-dimensional orbifolds, and predict that it should be true in general also. We generalize this question in the category of Lie groupoids, as orbifolds can be identified with a certain kind of Lie groupoids.

已知 Mn 中所有 i≠j 的超平面 {xi=xj} 的补集是非球面的，其中 M 是一个非球面的 2 维漫游体。这里我们考虑 M 是二维轨道的情况。我们证明了对于一类非球面二维球面来说，这个补集是非球面的，并预言它在一般情况下也应该是正确的。我们将这一问题推广到烈群范畴，因为轨道可以与某类烈群相提并论。

引用次数: 0

Sequences with increasing subsequence 具有递增子序列的序列

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2024-08-26 DOI: 10.1016/j.topol.2024.109049

Łukasz Mazurkiewicz, Szymon Żeberski

We study analytic and Borel subsets defined similarly to the old example of analytic complete set given by Luzin. Luzin's example, which is essentially a subset of the Baire space, is based on the natural partial order on naturals, i.e. division. It consists of sequences which contain increasing subsequence in given order.

We consider a variety of sets defined in a similar way. Some of them occurs to be Borel subsets of the Baire space, while others are analytic complete, hence not Borel.

In particular, we show that an analogon of Luzin example based on the natural linear order on rationals is analytic complete. We also characterize all countable linear orders having such property.

我们研究的解析子集和玻尔子集的定义与卢津给出的解析完全集的老例子类似。卢津的例子本质上是拜尔空间的一个子集，它基于自然数的自然偏序，即除法。它由按给定顺序包含递增子序列的序列组成。我们考虑了以类似方式定义的各种集合，其中有些集合是贝叶尔空间的贝叶尔子集，而另一些则是解析完全集，因此不是贝叶尔集。我们还描述了具有这种性质的所有可数线性阶的特征。

引用次数: 0

D-completion, well-filterification and sobrification D 级完井、滤井和净化

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2024-08-26 DOI: 10.1016/j.topol.2024.109050

Hualin Miao , Longchun Wang , Qingguo Li

In this paper, we study the D-completion, well-filterification and sobrification of a $T_{0}$ space. First, we present an example of a tapered closed set which is neither the closure of a directed set nor a closed KF-set. In 2020, Xu et al. asked whether closed RD-sets are exactly closed WD-sets for every $T_{0}$ space. This example also gives a negative answer to the above problem, since each tapered closed set is a closed WD-set. Second, we provide a direct characterization for the D-completion of a poset by using the notion of pre-C-compact elements. Finally, for a given $T_{0}$ space, we give some sufficient conditions which guarantee that each pair of its standard D-completion, standard well-filterification and standard sobrification agrees.

在本文中，我们研究了 T0 空间的 D-补全、井滤化和索布尔化。首先，我们举例说明既不是有向集的闭集也不是闭 KF 集的锥形闭集。2020 年，Xu 等人提出了这样一个问题：对于每个 T0 空间，闭 RD 集是否都是完全闭的 WD 集？这个例子也给出了上述问题的否定答案，因为每个锥形闭集都是一个闭 WD 集。其次，我们利用前 C-紧凑元素的概念，为正集的 D-补集提供了一个直接表征。最后，对于给定的 T0 空间，我们给出了一些充分条件，以保证其标准 D-补集、标准井滤波和标准净化的每一对都是一致的。

引用次数: 0

Dynamic properties of the dynamical system (FnK(X),FnK(f)) 动力系统 (FnK(X),FnK(f))的动态特性

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2024-08-22 DOI: 10.1016/j.topol.2024.109048

Franco Barragán , Anahí Rojas , Jesús F. Tenorio

<div>Let <math><mo>(</mo><mi>X</mi><mo>,</mo><mi>f</mi><mo>)</mo></math> be a dynamical system, where X is a nondegenerate continuum and f is a map. For any positive integer n, we consider the hyperspace <math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math> with the Vietoris topology. For <math><mi>n</mi><mo>></mo><mn>1</mn></math> and <math><mi>K</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math> the subset <math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>K</mi><mo>,</mo><mi>X</mi><mo>)</mo></math> of <math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math> is defined as the collection of elements of <math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math> containing K. We consider the quotient hyperspace <math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>K</mi></mrow></msubsup><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mi>⧸</mi><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>K</mi><mo>,</mo><mi>X</mi><mo>)</mo></math>, which is obtained from <math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math> by shrinking <math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>K</mi><mo>,</mo><mi>X</mi><mo>)</mo></math> to one point set. Furthermore, we consider the induced maps <math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>f</mi><mo>)</mo><mo>:</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mo>→</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math> and <math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>K</mi></mrow></msubsup><mo>(</mo><mi>f</mi><mo>)</mo><mo>:</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>K</mi></mrow></msubsup><mo>(</mo><mi>X</mi><mo>)</mo><mo>→</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>K</mi></mrow></msubsup><mo>(</mo><mi>X</mi><mo>)</mo></math>. In this paper, we introduce the dynamical system <math><mo>(</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>K</mi></mrow></msubsup><mo>(</mo><mi>X</mi><mo>)</mo><mo>,</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>K</mi></mrow></msubsup><mo>(</mo><mi>f</mi><mo>)</mo><mo>)</mo></math> and we study relationships between the conditions <math><mi>f</mi><mo>∈</mo><mi>M</mi></math>, <math><msub><mrow><mi>F</mi></mr

假设 (X,f) 是一个动力系统，其中 X 是一个非enerate 连续体，f 是一个映射。对于任意正整数 n，我们考虑具有 Vietoris 拓扑的超空间 Fn(X)。对于 n>1 和 K∈Fn(X)，Fn(X) 的子集 Fn(K,X) 被定义为 Fn(X) 中包含 K 的元素集合。我们考虑商超空间 FnK(X)=Fn(X)⧸Fn(K,X) ，它是通过将 Fn(K,X) 缩小到一个点集而从 Fn(X) 得到的。此外，我们还考虑了诱导映射 Fn(f):Fn(X)→Fn(X) 和 FnK(f):FnK(X)→FnK(X) 。本文引入动力系统 (FnK(X),FnK(f))，并研究条件 f∈M、Fn(f)∈M 和 FnK(f)∈M 之间的关系，其中 M 是以下几类映射之一：传递、混合、弱混合、完全传递、精确、阿金-奥斯兰德-纳加尔意义上的精确、阿金-奥斯兰德-纳加尔意义上的强传递、精确传递、完全精确、强精确传递、强积传递、轨道传递、德瓦尼混沌、不可还原、TT++、强传递和极强传递。

引用次数: 0

Compact subspaces of the space of separately continuous functions with the cross-uniform topology 具有交叉均匀拓扑的分别连续函数空间的紧凑子空间

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2024-08-22 DOI: 10.1016/j.topol.2024.109047

Oleksandr Maslyuchenko , Vadym Myronyk , Roman Ivasiuk

We consider two natural topologies on the space $S (X \times Y, Z)$ of all separately continuous functions defined on the product of two topological spaces X and Y and ranged into a topological or metric space Z. These topologies are the cross-open topology and the cross-uniform topology. We show that these topologies coincides if X and Y are pseudocompacts and Z is a metric space. We prove that a compact space K embeds into $S (X \times Y, Z)$ for infinite compacts X, Y and a metrizable space $Z \supseteq R$ if and only if the weight of K is less than the sharp cellularity of both spaces X and Y.

我们考虑了空间 S（X×Y,Z）上的两个自然拓扑，S（X×Y,Z）是定义在两个拓扑空间 X 和 Y 的乘积上的所有独立连续函数，并被置换到一个拓扑或度量空间 Z 中。我们证明，如果 X 和 Y 是伪紧凑且 Z 是度量空间，这些拓扑就会重合。我们证明，对于无限紧凑的 X、Y 和可元空间 Z⊇R，当且仅当 K 的权重小于 X 和 Y 两个空间的锐胞度时，紧凑空间 K 嵌入到 S(X×Y,Z)中。

引用次数: 0

The generalized metric property in strongly topological gyrogroups 强拓扑陀螺群中的广义度量属性

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2024-08-22 DOI: 10.1016/j.topol.2024.109046

Meng Bao, Xiaolan Liu

A topological gyrogroup is a gyrogroup endowed with a topology such that the binary operation is jointly continuous and the inverse mapping is also continuous. It is shown that a strongly topological gyrogroup is a q-space if and only if it is an M-space. Then a characterization about weakly feathered strongly topological gyrogroups is given, that is, a strongly topological gyrogroup G is weakly feathered if and only if it contains a compact strong subgyrogroup H such that the quotient space $G / H$ is submetrizable.

拓扑陀螺群是具有拓扑结构的陀螺群，其二元操作是连续的，反映射也是连续的。研究表明，强拓扑陀螺群是一个 q 空间，当且仅当它是一个 M 空间。然后给出了一个关于弱羽化强拓扑陀螺群的特征，即强拓扑陀螺群 G 是弱羽化的，当且仅当它包含一个紧凑的强子陀螺群 H，使得商空间 G/H 是可亚对称的。

引用次数: 0

Meeting, covering and Shelah's Revised GCH 会议、报道和谢拉的修订版 GCH

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2024-08-20 DOI: 10.1016/j.topol.2024.109044

Pierre Matet

We revisit the application of Shelah's Revised GCH Theorem [19] to diamond. We also formulate a generalization of the theorem and prove a small fragment of it. Finally we consider another application of the theorem, to covering numbers of the form $cov (-, -, -, ω)$ .

我们重温了谢拉的修正 GCH 定理 [19] 在金刚石中的应用。我们还提出了该定理的一般化，并证明了其中的一小部分。最后，我们考虑了该定理在 cov(-,-,-,ω) 形式的覆盖数上的另一种应用。

引用次数: 0

Splittings of tangles and spatial graphs 纠结和空间图的分裂

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2024-08-20 DOI: 10.1016/j.topol.2024.109042

Erica Flapan , Hugh Howards

Menasco proved that if G is a reduced, alternating, connected diagram of a link L and G is prime then L is prime. This surprising and important result has been generalized to other classes of links, as well as to tangles and spatial graphs. After exploring some issues with previous results, we obtain new splitting results for tangles and spatial graphs.

梅纳斯科证明，如果 G 是链接 L 的还原、交替、连通图，并且 G 是素数，那么 L 就是素数。这一惊人而重要的结果已被推广到其他类别的链接以及缠结和空间图。在探讨了之前结果的一些问题之后，我们得到了缠结图和空间图表的新分裂结果。

引用次数: 0

On the sequential topological complexity of group homomorphisms 论群同态的顺序拓扑复杂性

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2024-08-16 DOI: 10.1016/j.topol.2024.109045

Nursultan Kuanyshov

We define and develop a homotopy invariant notion for the sequential topological complexity of a map $f : X \to Y$ , denoted $T C_{r} (f)$ , that interacts with $T C_{r} (X)$ and $T C_{r} (Y)$ in the same way Jamie Scott's topological complexity map $T C (f)$ interacts with $T C (X)$ and $T C (Y)$ . Furthermore, we apply $T C_{r} (f)$ to studying group homomorphisms $ϕ : Γ \to Λ$ .

In addition, we give the characterization of cohomological dimension of group homomorphisms.

我们定义并发展了一个同调不变的概念，即映射 f:X→Y 的序列拓扑复杂性，记为 TCr(f)，它与 TCr(X) 和 TCr(Y) 交互作用的方式与杰米-斯科特的拓扑复杂性映射 TC(f) 与 TC(X) 和 TC(Y) 交互作用的方式相同。此外，我们还将 TCr(f) 应用于研究群同态 j:Γ→Λ。

引用次数: 0

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