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Pseudocompact versus countably compact in first countable spaces 第一可数空间中的伪紧与可数紧

IF 0.5 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2025-10-28 DOI: 10.1016/j.topol.2025.109650

István Juhász , Lajos Soukup , Zoltán Szentmiklóssy

The primary objective of this work is to construct spaces that are “pseudocompact but not countably compact”, abbreviated as P-NC, while endowing them with additional properties.

First, motivated by an old problem of van Douwen concerning first countable P-NC spaces with countable extent, we construct from CH a locally compact and locally countable first countable P-NC space with countable spread.

A space is deemed densely countably compact, denoted as DCC for brevity, if it possesses a dense, countably compact subspace. Moreover, a space qualifies as densely relatively countably compact, abbreviated as DRC, if it contains a dense subset D such that every infinite subset of D has an accumulation point in X.

A countably compact space is DCC, a DCC space is DRC, and a DRC space is evidently pseudocompact. The Tychonoff plank is a DCC space but is not countably compact. A Ψ-space belongs to the class of DRC spaces but is ¬DCC. Lastly, if

p \in ω^{⁎}

is not a P-point, then

T (p)

, representing the type of p in

ω^{⁎}

, constitutes a pseudocompact subspace of

ω^{⁎}

that is ¬DRC.

When considering a topological property denoted as Q, we define a space X as “R-hereditarily Q” if every regular closed subspace of X also possesses property Q. The Tychonoff plank and the Ψ-space are not R-hereditary examples for separating the above-mentioned properties. However, the aforementioned space

T (p)

is an R-hereditary example, albeit not being first countable.

In this paper we want to find (first countable) examples which separate these properties R-hereditarily. We have obtained the following result.

(1)
There is a R-hereditarily “DCC, but not countably compact” space.
(2)
If CH holds, then there is a R-hereditarily “DRC, but ¬DCC” space.
(3)
If $s = c$ , then there is a first countable, R-hereditarily “pseudocompact, but ¬DRC” space.

In contrast to (2), it is unknown whether a first countable, R-hereditarily “DRC, but ¬DCC” space X can exist.

本工作的主要目标是构建“伪紧但不可可数紧”的空间，缩写为P-NC，同时赋予它们额外的性质。首先，根据van Douwen关于可数扩展的首可数P-NC空间的一个老问题，我们从CH构造了一个具有可数扩展的局部紧且局部可数的首可数P-NC空间。如果一个空间具有一个密集的、可数紧的子空间，则认为它是密集可数紧的，为简洁起见，记为DCC。此外，如果一个空间包含一个稠密子集D，使得D的每一个无限子集在x中都有一个累加点，那么这个空间就是密集相对可数紧的，缩写为DRC。一个可数紧空间是DCC，一个DCC空间是DRC，一个DRC空间明显是伪紧的。Tychonoff木板是一个DCC空间，但不是可数的紧凑。Ψ-space属于DRC空间类，但属于rdcc。最后，如果p∈ω不是p点，则表示ω中p的类型的T(p)构成ω的伪紧子空间，该伪紧子空间为ω。当考虑用Q表示的拓扑性质时，如果X的每一个正则闭子空间也具有性质Q，我们定义空间X为“r -遗传性Q”。Tychonoff板和Ψ-space不是分离上述性质的r -遗传性例子。然而，前面提到的空间T(p)是一个r -遗传的例子，尽管不是第一可数的。在本文中，我们想找到（第一个可数的）例子来分离这些性质r -遗传。我们得到了以下结果。(1)存在一个r -遗传的“DCC，但不可数紧”空间。(2)如果CH成立，则存在一个r -遗传的“DRC，但不DCC”空间。(3)如果s=c，则存在一个第一可数的、r -遗传的“伪紧，但不DRC”空间。与(2)相反，未知是否存在第一可数的r -遗传的“DRC，但DCC”空间X。

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引用次数: 0

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 <span><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></span>, whenever <em>X</em> is a Hausdorff space, by showing that <span><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></span>, for every Hausdorff space <em>X</em>. Recently, Bella proved that if <em>X</em> is a regular <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-space, then <span><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></span> and he asked if the same inequality holds true for every Hausdorff space <em>X</em>. In this paper, for every Hausdorff space <em>X</em> we show that <span><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></span>, and for every Urysohn space <em>X</em>, we prove that <span><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></span>. Since for regular <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-spaces <em>X</em> we have <span><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></span>, and for every Urysohn space <em>X</em> we have <span><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></span> and <span><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

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