Topology and its Applications最新文献

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On the group action version of the Kuratowski-Mycielski theorem and invariant chaotic sets 库拉托夫斯基-米切尔斯基定理的群作用版本与不变混沌集

IF 0.5 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2025-10-28 DOI: 10.1016/j.topol.2025.109649

Jiuzhi Gao, Ziyu Huang

Let

(X, G)

be a dynamical system with X a perfect Polish space and G a countable group, and let

K (X)

denote the collection of all compact subsets of X. It is shown that if Q is a

G_{δ}

, hereditary subset of

K (X)

and

α_{Q} = {R_{n}^{Q}}_{n \in N}

is the coherent list on X associated with Q, then a group action version of Kuratowski-Mycielski theorem holds.

Meanwhile, we construct a non-trivial transitive system

(X, G)

with G a countable abelian group, such that there exist some special invariant chaotic sets in X. Specifically, there exists a G-invariant, n-

δ_{n}

-scrambled, uniformly chaotic set in

Σ_{2}

设（X，G）是一个动力系统，其中X是完美波兰空间，G是可数群，K(X)表示X的所有紧子集的集合。证明了如果Q是一个Gδ， K(X)的遗传子集和αQ={RnQ}n∈n是X上与Q相关的相干表，则群作用版的Kuratowski-Mycielski定理成立。同时，我们构造了一个非平凡的传递系统（X，G），其中G是可数阿贝尔群，使得X中存在一些特殊的不变混沌集，其中在Σ2中存在一个G不变的n-δn-置乱的一致混沌集。

引用次数: 0

Star covering properties of products of subspaces of ordinals 序数子空间积的星覆盖性质

IF 0.5 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2025-10-27 DOI: 10.1016/j.topol.2025.109648

Yanhui Huang

In this paper, we discuss the relationships among

ω_{1}

-compactness, star countability, star Lindelöfness, star almost Lindelöfness and star weakly Lindelöfness in different spaces. We mainly give the following:

(1)
For a subspace X of an ordinal, X is star weakly Lindelöf if and only if it is $ω_{1}$ -compact.
(2)
For subspaces A and B of an ordinal, $A \times B$ is star weakly Lindelöf if and only if it is $ω_{1}$ -compact.
(3)
For a subspace X of $ω_{1}^{2}$ , X is star weakly Lindelöf if and only if it is $ω_{1}$ -compact.

本文讨论了不同空间中ω1紧性、星可数性、星Lindelöfness、星几乎Lindelöfness和星弱Lindelöfness之间的关系。我们主要给出以下结论：(1)对于序数的子空间X，当且仅当它是ω1紧的，X是弱星形Lindelöf。(2)对于序数的子空间A和子空间B， A×B是弱星型Lindelöf当且仅当它是ω - 1紧的。(3)对于ω12的子空间X，当且仅当它是ω1紧时，X是弱星形Lindelöf。

引用次数: 0

Distinguished dense Cp-subspaces 区分稠密的cp子空间

IF 0.5 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2025-10-27 DOI: 10.1016/j.topol.2025.109647

J.C. Ferrando , J. Ka̧kol

Let

C_{p} (X)

be the linear space of real-valued continuous functions with the pointwise topology. It is known that a Tychonoff space X is a Δ-space if and only if the locally convex space

C_{p} (X)

is distinguished. It has been recently shown that if there is a continuous linear surjection from

C_{p} (X)

onto

C_{p} (Y)

and X is a Δ-space, Y is also a Δ-space. Here we investigate under what conditions the presence of a dense distinguished subspace E in

C_{p} (X)

leads X to be a Δ-space. We also produce a class of spaces

X \notin Δ

for which

C_{p} (X)

contains a distinguished dense subspace.

设Cp(X)为具有点向拓扑的实值连续函数的线性空间。已知Tychonoff空间X是Δ-space当且仅当局部凸空间Cp(X)被区分。最近已经证明，如果存在从Cp(X)到Cp(Y)的连续线性抛射，并且X是Δ-space， Y也是Δ-space。在这里，我们研究在什么条件下，在Cp(X)中存在一个稠密的可分辨子空间E导致X是Δ-space。我们也得到了一类空间X∈Δ，其中Cp(X)包含一个可分辨的稠密子空间。

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引用次数: 0

Hyperbolic handlebody complements in 3-manifolds 3流形中的双曲柄体补

IF 0.5 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2025-10-24 DOI: 10.1016/j.topol.2025.109642

Colin Adams , Francisco Gomez-Paz , Jiachen Kang , Lukas Krause , Gregory Li , Chloe Marple , Ziwei Tan

Let

M_{0}

be a compact and orientable 3-manifold. After capping off spherical boundaries with balls and removing any torus boundaries, we prove that the resulting manifold M contains handlebodies of arbitrary genus such that the closure of their complement is hyperbolic. We then extend the octahedral decomposition to obtain bounds on volume for some of these handlebody complements.

设M0是一个紧致可定向的3流形。在用球对球面边界进行封顶并去掉环面边界后，我们证明了得到的流形M包含任意属的柄体，使得它们的补的闭包是双曲的。然后对八面体分解进行扩展，得到其中一些柄体补体的体积边界。

引用次数: 0

Closed left ideal decompositions of βG ∖ G and wandering points βG ∈ G与游荡点的闭左理想分解

IF 0.5 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2025-10-22 DOI: 10.1016/j.topol.2025.109646

Valentin Keyantuo , Yevhen Zelenyuk

Let G be a countably infinite discrete group, let βG be the Stone-Čech compactification of G, and let

G^{⁎} = β G ∖ G

. For every closed left ideal

X \subseteq G^{⁎}

of βG, there is a finest decomposition

D (X)

of X into closed left ideals of βG with the property that the corresponding quotient space of X is Hausdorff. It is known that

D (G^{⁎})

is nontrivial, and in fact

| D (G^{⁎}) | = 2^{c}

, and for some

X \in D (G^{⁎})

D (X)

is nontrivial. We show that it is consistent with ZFC that, if G can be embedded algebraically into a compact group, then for every

X \in D (G)

D (X)

is nontrivial.

设G是一个可数无限离散群，设βG是G的石头紧化-Čech，设G→=βG→G。对于βG的每一个闭左理想X，存在X的一个最优分解D(X)成βG的闭左理想，其性质为X对应的商空间为Hausdorff。已知D(G)是非平凡的，事实上|D(G)|=2c，并且对于某些X∈D(G)， D(X)是非平凡的。证明了如果G可以代数嵌入到紧群中，则对于每一个X∈D(G)， D(X)是非平凡的，这与ZFC是一致的。

引用次数: 0

Knotted surfaces, homological norm and extendable subgroup 结曲面、同调范数与可扩展子群

IF 0.5 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2025-10-21 DOI: 10.1016/j.topol.2025.109644

Qiling Liu

We prove that for an arbitrary g, there is a surface K of genus g embedded in

S^{4}

, which has finitely many extendable self-homeomorphisms' action on

H_{1} (K, Z)

, by defining a norm on

H_{1} (K, Z)

and proving its additivity.

通过定义H1（K，Z）上的范数并证明其可加性，证明了对于任意g，存在一个嵌入在S4中的g属曲面K，它在H1（K，Z）上具有有限多个可扩展自同胚的作用。

引用次数: 0

Integral and rational cohomologies of some generalized Dold manifolds 一些广义Dold流形的积分与有理上同调

IF 0.5 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2025-10-21 DOI: 10.1016/j.topol.2025.109645

S. Kavitha , V. Renukadevi

Let

σ : X \to X

be an involution on a manifold X with non empty fixed point set. The generalized Dold manifold

D (m, X)

is a space obtained as the quotient of

S^{m} \times X

by the action of

Z_{2}

defined by the fixed point free involution

(a, x) \mapsto (- a, σ (x))

. When X is either a complex Grassmannian manifold

G_{k} (C^{n})

or a complete flag manifold

F ℓ (n)

, we obtain the integral cohomology group of the generalized Dold manifolds

D (m, G_{k} (C^{n}))

and

D (m, F ℓ (n))

from the well known cell structures of complex Grassmannian and complete flag manifold. Also, we give the rational cohomology of the generalized Dold manifolds

D (m, G_{k} (C^{n}))

and

D (m, F ℓ (n))

设σ:X→X是流形X上具有非空不动点集的对合。广义Dold流形D（m，X）是由不动点自由对合（a， X）∑(- a,σ(X))定义的Z2作用下得到的作为Sm×X商的空间。当X是复格拉斯曼流形Gk（Cn）或完全标志流形F r (n)时，我们从已知的复格拉斯曼流形和完全标志流形的胞结构中得到广义Dold流形D（m,Gk(Cn)）和D（m,F r (n)）的积分上同调群。同时给出了广义Dold流形D（m,Gk(Cn)）和D（m,F (n)）的有理上同调。

引用次数: 0

Strengthening of de Groot's, Spadaro's, and Bella's inequalities for Urysohn spaces Urysohn空间中de Groot、Spadaro和Bella不等式的强化

IF 0.5 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2025-10-21 DOI: 10.1016/j.topol.2025.109643

Ivan S. Gotchev

<div><div>In 2011, Spadaro improved de Groot's inequality <math><mo>|</mo><mi>X</mi><mo>|</mo><mo>≤</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>h</mi><mi>L</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msup></math>, whenever X is a Hausdorff space, by showing that <math><mo>|</mo><mi>X</mi><mo>|</mo><mo>≤</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>L</mi><mo>(</mo><mi>X</mi><mo>)</mo><mi>F</mi><mo>(</mo><mi>X</mi><mo>)</mo><mi>ψ</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msup></math>, for every Hausdorff space X. Recently, Bella proved that if X is a regular <math><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math>-space, then <math><mo>|</mo><mi>X</mi><mo>|</mo><mo>≤</mo><mi>h</mi><mi>L</mi><msup><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mrow><mi>F</mi><mo>(</mo><mi>X</mi><mo>)</mo><mi>ψ</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msup></math> and he asked if the same inequality holds true for every Hausdorff space X. In this paper, for every Hausdorff space X we show that <math><mo>|</mo><mi>X</mi><mo>|</mo><mo>≤</mo><mi>h</mi><mi>L</mi><msup><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mrow><mi>F</mi><mo>(</mo><mi>X</mi><mo>)</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msup></math>, and for every Urysohn space X, we prove that <math><mo>|</mo><mi>X</mi><mo>|</mo><mo>≤</mo><mi>h</mi><mi>L</mi><msup><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>θ</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msup></math>. Since for regular <math><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math>-spaces X we have <math><msub><mrow><mi>ψ</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><mi>ψ</mi><mo>(</mo><mi>X</mi><mo>)</mo></math>, and for every Urysohn space X we have <math><mi>h</mi><mi>L</mi><msup><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mrow><mi>F</mi><mo>(</mo><mi>X</mi><mo>)</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msup><mo>≤</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>L</mi><mo>(</mo><mi>X</mi><mo>)</mo><mi>F</mi><mo>(</mo><mi>X</mi><mo>)</mo><mi>ψ</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msup></math> and <math><mi>h</mi><mi>L</mi><msup><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>θ</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><msub><mrow><mi>ψ</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msup><mo>≤</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>L</mi><mo>(</mo><mi>X</mi><mo

在2011年，Spadaro改进了de Groot的不等式|X|≤2hL(X)，当X是Hausdorff空间时，通过证明|X|≤2L(X)F(X)ψ(X)，对于每一个Hausdorff空间X。最近，Bella证明了如果X是正则t - 1空间，那么|X|≤hL(X)F(X)ψ(X)，并且他问是否同样的不等式对于每一个Hausdorff空间X成立。本文证明了|X|≤hL(X)F(X)ψ(X)，对于每一个Urysohn空间X，我们证明了|X|≤hL(X)Fθ(X)ψ4(X)。由于对于正则t1 -空间X我们有ψ3(X)=ψ4(X)=ψ(X)，并且对于每一个Urysohn空间X我们有hL(X)F(X)ψ3(X)≤2L(X)F(X)ψ(X)和hL(X)Fθ(X)ψ4(X)≤2L(X)F(X)ψ(X)，我们的结果改进了de Groot’s, Spadaro’s和Bella’s关于Urysohn空间的不等式。

引用次数: 0

Weird R-factorizable groups 奇怪的r可分解群

IF 0.5 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2025-10-20 DOI: 10.1016/j.topol.2025.109640

Evgenii Reznichenko, Ol'ga Sipacheva

The problem of the existence of non-pseudo-

ℵ_{1}

-compact

R

-factorizable groups is studied. It is proved that any such group is submetrizable and has weight larger than

ω_{1}

. Closely related results concerning the

R

-factorizability of products of topological groups and spaces are also obtained (a product

X \times Y

of topological spaces is said to be

R

-factorizable if any continuous function

X \times Y \to R

factors through a product of maps from X and Y to second-countable spaces). In particular, it is proved that the square

G \times G

of a topological group G is

R

-factorizable as a group if and only if it is

R

-factorizable as a product of spaces, in which case G is pseudo-

ℵ_{1}

-compact. It is also proved that if the product of a space X and an uncountable discrete space is

R

-factorizable, then

X^{ω}

is hereditarily separable and hereditarily Lindelöf.

研究了非伪1-紧r -可分解群的存在性问题。证明了这类群是可次幂的，其权值大于ω1。我们还得到了拓扑群与空间乘积的R可分解性的密切相关的结果（如果任何连续函数X×Y→R因子通过X和Y映射到次可数空间的乘积，则拓扑空间的乘积X×Y是R可分解的）。特别地，证明了拓扑群G的平方G×G作为群是可r分解的当且仅当它作为空间的乘积是可r分解的，在这种情况下G是伪- α -紧的。还证明了如果空间X与不可数离散空间的积是r可分解的，则Xω是遗传可分的，并且遗传Lindelöf。

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引用次数: 0

Maximal ω1-compactness 最大ω1-compactness

IF 0.5 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2025-10-17 DOI: 10.1016/j.topol.2025.109637

Ofelia T. Alas , L. Enrique Gutiérrez-Domínguez , Richard G. Wilson

A space is said to be

ω_{1}

-compact if every closed and discrete subset is countable. In this paper we study spaces whose topologies are maximal with respect to being

ω_{1}

-compact and we give characterisations of this property in a number of special cases. We also consider the preservation of

ω_{1}

-compactness in products.

如果一个空间的每一个封闭的离散子集都是可数的，我们就说这个空间是ω1紧的。本文研究了拓扑在ω -紧性方面是极大的空间，并在一些特殊情况下给出了这一性质的表征。我们还考虑了产品中ω - 1紧性的保存性。

引用次数: 0

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