Topology and its Applications最新文献

英文中文

Knots in RP3 RP3中的结

IF 0.5 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2025-11-05 DOI: 10.1016/j.topol.2025.109656

Louis H. Kauffman , Rama Mishra , Visakh Narayanan

This paper studies knots in three dimensional projective space. Techniques in virtual knot theory are applied to obtain a Jones polynomial for projective links and it is shown that this is equivalent to the Jones polynomial defined by Drobotukhina. A Khovanov homology theory for projective knots is constructed by using virtual Khovanov homology and the virtual Rasmussen invariant of Dye, Kaestner, and Kauffman. This homology theory is compared with the Khovanov theory developed by Manolescu and Willis for projective knots. It is shown that these theories are essentially equivalent, giving new viewpoints for both methods. The paper ends with problems about these approaches and an example of multiple projectivizations of the figure-8 knot whose equivalence is unknown at this time.

本文研究了三维射影空间中的结。应用虚结理论中的技术，得到了投影连杆的Jones多项式，并证明了它与Drobotukhina定义的Jones多项式是等价的。利用Dye、Kaestner和Kauffman的虚Khovanov同调和虚Rasmussen不变量，构造了射影节的Khovanov同调理论。将该同调理论与Manolescu和Willis提出的关于射角结的Khovanov理论进行了比较。结果表明，这两种理论在本质上是等价的，为两种方法提供了新的观点。文章最后给出了这些方法的问题，并给出了8字形结的多重投影的一个例子，这个例子的等价性目前是未知的。

引用次数: 0

Constructions of and bounds on the toric mosaic number 复向镶嵌数的构造及界

IF 0.5 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2025-10-31 DOI: 10.1016/j.topol.2025.109657

Kendall Heiney , Margaret Kipe , Samantha Pezzimenti , Kaelyn Pontes , Lực Ta

Knot mosaics were introduced by Kauffman and Lomonaco in the context of quantum knots, but have since been studied for their own right. A classical knot mosaic is formed on a square grid. In this work, we identify opposite edges of the square to form mosaics on the surface of a torus. We provide two algorithms for efficiently constructing toric mosaics of torus knots, providing upper bounds for the toric mosaic number. Using these results and a computer search, we provide a census of known toric mosaic numbers.

结镶嵌是由Kauffman和Lomonaco在量子结的背景下引入的，但后来被研究了自己的权利。一个经典的花结镶嵌在方形网格上。在这个作品中，我们识别出正方形的相对边缘，在一个环面的表面上形成马赛克。我们提供了两种有效构造环面结点的环面镶嵌的算法，并给出了环面镶嵌数的上界。利用这些结果和计算机搜索，我们提供了已知环面马赛克数的普查。

引用次数: 0

Spaces of countable type with hereditarily Baire Vietoris hyperspace 具有遗传Baire Vietoris超空间的可数型空间

IF 0.5 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2025-10-31 DOI: 10.1016/j.topol.2025.109655

Francis Jordan

We extend the characterization of the hereditary Baireness of the Vietoris hyperspace of separable metric spaces given by Gartside, Medini, and Zdomskyy [3] to the class of regular topological spaces of countable type. A theorem of Bouziad, Holá, and Zsilinszky [2] is also extended in a similar way.

本文将Gartside， Medini， Zdomskyy[3]给出的可分度量空间的Vietoris超空间的遗传bairenness的刻画推广到可数型正则拓扑空间。Bouziad, hol和zsilinsky[2]的定理也以类似的方式得到了推广。

引用次数: 0

Indecomposability of group actions 群体行为的不可分解性

IF 0.5 4区数学 Q3 MATHEMATICS

Topology and its Applications

Pub Date : 2025-10-30 DOI: 10.1016/j.topol.2025.109652

Ashwani K B, Ali Akbar K

A chaotic group action is a nonminimal, topologically transitive continuous group action with dense periodic points. In this paper, we discuss indecomposability for a continuous group action and prove that indecomposability is an equivalent definition of topological transitivity. Moreover, we prove that any infinite compact subset of the real line having a chaotic group action is homeomorphic to the middle third Cantor set.

混沌群作用是具有密集周期点的非极小、拓扑可传递的连续群作用。本文讨论了连续群作用的不可分解性，并证明了不可分解性是拓扑传递性的等价定义。此外，我们证明了具有混沌群作用的实线的无限紧子集与中三分之一康托集是同纯的。

引用次数: 0

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。

{"title":"Pseudocompact versus countably compact in first countable spaces","authors":"István Juhász , Lajos Soukup , Zoltán Szentmiklóssy","doi":"10.1016/j.topol.2025.109650","DOIUrl":"10.1016/j.topol.2025.109650","url":null,"abstract":"<div><div>The primary objective of this work is to construct spaces that are “<em>pseudocompact but not countably compact</em>”, abbreviated as P-NC, while endowing them with additional properties.</div><div>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.</div><div>A space is deemed <em>densely countably compact</em>, denoted as DCC for brevity, if it possesses a dense, countably compact subspace. Moreover, a space qualifies as <em>densely relatively countably compact</em>, abbreviated as DRC, if it contains a dense subset <em>D</em> such that every infinite subset of <em>D</em> has an accumulation point in <em>X</em>.</div><div>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 <span><math><mi>p</mi><mo>∈</mo><msup><mrow><mi>ω</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> is not a P-point, then <span><math><mi>T</mi><mo>(</mo><mi>p</mi><mo>)</mo></math></span>, representing the type of <em>p</em> in <span><math><msup><mrow><mi>ω</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>, constitutes a pseudocompact subspace of <span><math><msup><mrow><mi>ω</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> that is ¬DRC.</div><div>When considering a topological property denoted as <em>Q</em>, we define a space <em>X</em> as “<em>R-hereditarily Q</em>” if every regular closed subspace of <em>X</em> also possesses property <em>Q</em>. The Tychonoff plank and the Ψ-space are not R-hereditary examples for separating the above-mentioned properties. However, the aforementioned space <span><math><mi>T</mi><mo>(</mo><mi>p</mi><mo>)</mo></math></span> is an R-hereditary example, albeit not being first countable.</div><div>In this paper we want to find (first countable) examples which separate these properties R-hereditarily. We have obtained the following result.<ul><li><span>(1)</span><span><div>There is a R-hereditarily “DCC, but not countably compact” space.</div></span></li><li><span>(2)</span><span><div>If CH holds, then there is a R-hereditarily “DRC, but ¬DCC” space.</div></span></li><li><span>(3)</span><span><div>If <span><math><mi>s</mi><mo>=</mo><mi>c</mi></math></span>, then there is a first countable, R-hereditarily “pseudocompact, but ¬DRC” space.</div></span></li></ul> In contrast to (2), it is unknown whether a first countable, R-hereditarily “DRC, but ¬DCC” space <em>X</em> can exist.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109650"},"PeriodicalIF":0.5,"publicationDate":"2025-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145466896","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

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

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