Topology and its Applications最新文献

英文中文

Weakly chained spaces 弱链空间

IF 0.5 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2025-10-30 DOI: 10.1016/j.topol.2025.109651

Conrad Plaut

We introduce “weakly chained spaces”, which in the metric case can be defined in a single paragraph using only the definition of “metric space”. Using this simple notion we more or less completely resolve the question of when a metrizable space X has a generalized universal covering map, which we call the uniform universal cover (UU-cover): if and only if it is weakly chained. These concepts are defined for uniform spaces, but one may extend all results to metrizable topological spaces via the fine uniformity, and we describe the relationship between this work and that of Fischer-Zastrow on generalized universal covers. We also show that the UU-cover has the analogous properties to those of the traditional universal cover: universal, lifting, uniqueness and functorial.

One of our main results concerns conditions under which an inverse limit of metric spaces is weakly chained. This theorem, in turn, has applications (in another paper) to boundaries of geodesically complete, co-compact, proper CAT(0) spaces, which may be regarded as inverse limits of the (weakly chained) metric spheres at a basepoint.

我们引入了“弱链空间”，它在度量情况下可以只用“度量空间”的定义在单个段落中定义。利用这个简单的概念，我们或多或少地解决了当一个可度量空间X有一个广义全称覆盖映射的问题，我们称之为一致全称覆盖（uu -盖）：当且仅当它是弱链的。这些概念是在一致空间下定义的，但我们可以将所有结果通过精细均匀性推广到可度量的拓扑空间，并描述了这一工作与Fischer-Zastrow关于广义上覆盖的研究之间的关系。我们还证明了uu -盖与传统的万能盖具有类似的性质：通用性、提升性、唯一性和泛函性。我们的一个主要结果是关于度量空间的逆极限是弱链的条件。反过来，这个定理也应用于测地线完备、共紧、固有CAT(0)空间的边界（在另一篇论文中），这些空间可以看作是（弱链）度量球在基点处的逆极限。

引用次数: 0

Strongly well-filtered spaces and strong d-spaces 强良好过滤空间和强d空间

IF 0.5 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2025-10-30 DOI: 10.1016/j.topol.2025.109654

Xiaoquan Xu

The main purpose of this paper is to reveal some finer links between d-spaces and

T_{2}

-spaces by introducing and studying a new class of

T_{0}

-spaces — strongly well-filtered spaces. The relationships among

T_{2}

-spaces,

T_{1}

-spaces, sober spaces, (strongly) well-filtered spaces and (strong) d-spaces are discussed. It is shown that if

m a x (A) \neq \emptyset

and

↓ (A \cap K)

is closed for any nonempty closed set A and saturated compact set K of a

T_{0}

-space X, then X is strongly well-filtered. An unexpected result is proved which states that for any poset P, the Scott space ΣP is a strong d-space iff it is strongly well-filtered. So the Scott space of a complete lattice is always strongly well-filtered. Some basic properties of strongly well-filtered spaces are investigated. It is shown that the strong well-filteredness is closed-hereditary and saturated-hereditary, and every retract of a strongly well-filtered space is strongly well-filtered. We give two Scott spaces which are strongly well-filtered and an R-space but their product space is not a strong d-space. This answers an open question posed by Lawson and Xu. Hence the category S-

{Top}_{w}

of strongly well-filtered spaces and continuous mappings is not reflective in the category

{Top}_{0}

T_{0}

-spaces and continuous mappings. Finally, we investigate the conditions under which the Smyth power space and Scott power space of a

T_{0}

-space is strongly well-filtered. Several such conditions are given.

本文的主要目的是通过引入和研究一类新的t0空间-强良好过滤空间来揭示d空间和t2空间之间的一些更精细的联系。讨论了t2 -空间、t1 -空间、清醒空间、（强）良滤空间和（强）d-空间之间的关系。证明了对于t0空间X的任何非空闭集A和饱和紧集K，如果max(A)≠∅且↓（A∩K）是闭的，则X是强滤好的。证明了一个意想不到的结果，该结果表明，对于任何偏置P， Scott空间ΣP是一个强d空间，如果它是强良好过滤的。所以完全晶格的斯科特空间总是强滤好的。研究了强良滤空间的一些基本性质。证明了强滤滤性是封闭遗传的和饱和遗传的，并且一个强滤滤空间的每一个缩回都是强滤滤的。我们给出了两个强过滤的Scott空间和一个r空间但它们的乘积空间不是强d空间。这回答了劳森和徐提出的一个开放性问题。因此强良过滤空间和连续映射的范畴S-Topw在t -空间和连续映射的范畴Top0中不反映。最后，我们研究了t0空间的Smyth幂空间和Scott幂空间是强良好滤波的条件。给出了几个这样的条件。

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引用次数: 0

On local compactness, pseudocompactness, and homogeneity 关于局部紧性、伪紧性和齐性

IF 0.5 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2025-10-28 DOI: 10.1016/j.topol.2025.109653

Nathan Carlson

We give a new bound for the cardinality of a Tychonoff homogeneous space using cozero sets. This leads to improved cardinal inequalities for compact homogeneous spaces that generalize to the locally compact setting. In this connection it is also shown that

w (X) \leq n w {(X)}^{p c t (X)}

for any Hausdorff space X, where

p c t (X)

is the point-wise compactness type of X. This extends Arhangel′skiĭ's result that

w (X) = n w (X)

when X is compact Hausdorff. In addition pseudocompactness is investigated in connection with homogeneity. Among other results, we show that if X is a ccc locally compact noncompact space such that the one-point compactification of X is homogeneous and has character

c

, then X is pseudocompact. It follows that if X is either

{[0, 1]}^{c}

2^{c}

and

p \in X

then

X ﹨ {p}

is pseudocompact.

利用余零集给出了Tychonoff齐次空间的基性的一个新界。这导致改进的基数不等式的紧齐次空间，推广到局部紧设置。由此还证明了对于任意Hausdorff空间X， w(X)≤nw(X)pct(X)，其中pct(X)是X的点向紧性类型。这推广了Arhangel’ski’的结论，即当X是紧Hausdorff时w(X)=nw(X)。此外，还研究了赝紧性与均匀性的关系。在其他结果中，我们证明了如果X是ccc局部紧非紧空间，使得X的一点紧化齐次且具有特征c，则X是伪紧的。因此，如果X是[0,1]c或2c，且p∈X，则X{p}是伪紧的。

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引用次数: 0

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

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