Pub Date : 2024-11-12DOI: 10.1016/j.topol.2024.109133
Anton E. Lipin
A subset A of a topological space X is called relatively functionally countable (RFC) in X, if for each continuous function the set is countable. We prove that all RFC subsets of a product are countable, assuming that spaces are Tychonoff and all RFC subsets of every are countable. In particular, in a metrizable space every RFC subset is countable.
The main tool in the proof is the following result: for every Tychonoff space X and any countable set there is a continuous function such that the restriction of f to is injective.
如果对于每个连续函数 f:X→R 的集合 f[A] 是可数的，那么拓扑空间 X 的子集 A 称为 X 中的相对函数可数（RFC）。我们假定空间 Xn 是 Tychonoff 的，且每个 Xn 的所有 RFC 子集都是可数的，从而证明乘积 ∏n∈ωXn 的所有 RFC 子集都是可数的。证明的主要工具是下面的结果：对于每一个Tychonoff空间X和任何可数集Q⊆X，有一个连续函数f:Xω→R2，使得f对Qω的限制是注入的。
{"title":"Relatively functionally countable subsets of products","authors":"Anton E. Lipin","doi":"10.1016/j.topol.2024.109133","DOIUrl":"10.1016/j.topol.2024.109133","url":null,"abstract":"<div><div>A subset <em>A</em> of a topological space <em>X</em> is called <em>relatively functionally countable</em> (<em>RFC</em>) in <em>X</em>, if for each continuous function <span><math><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>R</mi></math></span> the set <span><math><mi>f</mi><mo>[</mo><mi>A</mi><mo>]</mo></math></span> is countable. We prove that all RFC subsets of a product <span><math><munder><mo>∏</mo><mrow><mi>n</mi><mo>∈</mo><mi>ω</mi></mrow></munder><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> are countable, assuming that spaces <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> are Tychonoff and all RFC subsets of every <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> are countable. In particular, in a metrizable space every RFC subset is countable.</div><div>The main tool in the proof is the following result: for every Tychonoff space <em>X</em> and any countable set <span><math><mi>Q</mi><mo>⊆</mo><mi>X</mi></math></span> there is a continuous function <span><math><mi>f</mi><mo>:</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>ω</mi></mrow></msup><mo>→</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> such that the restriction of <em>f</em> to <span><math><msup><mrow><mi>Q</mi></mrow><mrow><mi>ω</mi></mrow></msup></math></span> is injective.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"359 ","pages":"Article 109133"},"PeriodicalIF":0.6,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142654964","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}
Pub Date : 2024-11-09DOI: 10.1016/j.topol.2024.109134
Marta Kwela, Jacek Tryba
In the article we consider Marczewski-Burstin countably representable (in short: ) ideals. We propose a concept of extendability to ideals and provide some of its properties like the fact that it lies between the notions of ω-+-diagonalizability and countable separability. We also answer the question posed in [Topology Appl. 248 (2018), 149–163], by showing that the ideal is not .
{"title":"Extendability to Marczewski-Burstin countably representable ideals","authors":"Marta Kwela, Jacek Tryba","doi":"10.1016/j.topol.2024.109134","DOIUrl":"10.1016/j.topol.2024.109134","url":null,"abstract":"<div><div>In the article we consider Marczewski-Burstin countably representable (in short: <span><math><mi>MBC</mi></math></span>) ideals. We propose a concept of extendability to <span><math><mi>MBC</mi></math></span> ideals and provide some of its properties like the fact that it lies between the notions of <em>ω</em>-+-diagonalizability and countable separability. We also answer the question posed in [Topology Appl. 248 (2018), 149–163], by showing that the ideal <span><math><msub><mrow><mi>J</mi></mrow><mrow><mi>c</mi></mrow></msub></math></span> is not <span><math><mi>MBC</mi></math></span>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"359 ","pages":"Article 109134"},"PeriodicalIF":0.6,"publicationDate":"2024-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142654965","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}
Pub Date : 2024-11-08DOI: 10.1016/j.topol.2024.109132
John E. Porter
We revisit monotonically semi-neighborhood refining (MSNR) spaces which were introduced by Stares in 1996. MSNR spaces are shown to be lob-spaces with well-ordered (F). The relationships between MSNR spaces with other monotone covering properties are also explored. We show the existence of MSNR spaces that do not posses a monotone locally-finite refining operator and spaces with a monotone locally-finite refining operator that are not MSNR answering a question of Popvassilev and Porter. Compact MSNR spaces may not be metrizable in general, but compact MSNR LOTS are. GO-spaces whose underlying LOTS has a σ-closed-discrete dense subset are shown to have a monotone star-finite refining operator.
{"title":"MSNR spaces revisited","authors":"John E. Porter","doi":"10.1016/j.topol.2024.109132","DOIUrl":"10.1016/j.topol.2024.109132","url":null,"abstract":"<div><div>We revisit monotonically semi-neighborhood refining (MSNR) spaces which were introduced by Stares in 1996. MSNR spaces are shown to be lob-spaces with well-ordered (F). The relationships between MSNR spaces with other monotone covering properties are also explored. We show the existence of MSNR spaces that do not posses a monotone locally-finite refining operator and spaces with a monotone locally-finite refining operator that are not MSNR answering a question of Popvassilev and Porter. Compact MSNR spaces may not be metrizable in general, but compact MSNR LOTS are. GO-spaces whose underlying LOTS has a <em>σ</em>-closed-discrete dense subset are shown to have a monotone star-finite refining operator.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"358 ","pages":"Article 109132"},"PeriodicalIF":0.6,"publicationDate":"2024-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142656836","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}
Pub Date : 2024-11-06DOI: 10.1016/j.topol.2024.109129
Heng Zhang , Wenfei Xi , Yaoqiang Wu , Hongling Li
A topological group G is called -factorizable (resp. -factorizable) if every continuous real-valued function on G admits a factorization via a continuous homomorphism onto a topological group H with (resp. a first-countable group). The first purpose of this article is to discuss some characterizations of -factorizable groups. It is shown that a topological group G is -factorizable if and only if every continuous real-valued function on G is -uniformly continuous, if and only if for every cozero-set U of G, there exists a -subgroup N of G such that . Sufficient conditions on the -factorizable group G to be -factorizable are that G is τ-fine and τ-steady for a cardinal τ.
如果拓扑群 G 上的每个连续实值函数都可以通过连续同态因式分解到拓扑群 H 上，且ψ(H)≤ω（或第一可数群），那么这个拓扑群 G 称为Ψω可因式分解群（或 M 可因式分解群）。本文的第一个目的是讨论Ψω可因子群的一些特征。本文指出，当且仅当对于 G 的每一个零集 U，存在一个 G 的 Gδ 子群 N，使得 UN=U 时，G 上的每一个连续实值函数都是 Gδ-uniformly 连续函数，拓扑群 G 才是Ψω-可因子群。Ψω-可因式化群 G 成为 M-可因式化群 G 的充分条件是 G 是τ-精细的，并且对于一个心数 τ 是τ-稳定的。
{"title":"On Ψω-factorizable groups","authors":"Heng Zhang , Wenfei Xi , Yaoqiang Wu , Hongling Li","doi":"10.1016/j.topol.2024.109129","DOIUrl":"10.1016/j.topol.2024.109129","url":null,"abstract":"<div><div>A topological group <em>G</em> is called <span><math><msub><mrow><mi>Ψ</mi></mrow><mrow><mi>ω</mi></mrow></msub></math></span>-factorizable (resp. <span><math><mi>M</mi></math></span>-factorizable) if every continuous real-valued function on <em>G</em> admits a factorization via a continuous homomorphism onto a topological group <em>H</em> with <span><math><mi>ψ</mi><mo>(</mo><mi>H</mi><mo>)</mo><mo>≤</mo><mi>ω</mi></math></span> (resp. a first-countable group). The first purpose of this article is to discuss some characterizations of <span><math><msub><mrow><mi>Ψ</mi></mrow><mrow><mi>ω</mi></mrow></msub></math></span>-factorizable groups. It is shown that a topological group <em>G</em> is <span><math><msub><mrow><mi>Ψ</mi></mrow><mrow><mi>ω</mi></mrow></msub></math></span>-factorizable if and only if every continuous real-valued function on <em>G</em> is <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>δ</mi></mrow></msub></math></span>-uniformly continuous, if and only if for every cozero-set <em>U</em> of <em>G</em>, there exists a <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>δ</mi></mrow></msub></math></span>-subgroup <em>N</em> of <em>G</em> such that <span><math><mi>U</mi><mi>N</mi><mo>=</mo><mi>U</mi></math></span>. Sufficient conditions on the <span><math><msub><mrow><mi>Ψ</mi></mrow><mrow><mi>ω</mi></mrow></msub></math></span>-factorizable group <em>G</em> to be <span><math><mi>M</mi></math></span>-factorizable are that <em>G</em> is <em>τ</em>-fine and <em>τ</em>-steady for a cardinal <em>τ</em>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"358 ","pages":"Article 109129"},"PeriodicalIF":0.6,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142656835","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}
Pub Date : 2024-11-05DOI: 10.1016/j.topol.2024.109131
Taras Radul
We introduce a functor of functionals that preserve the maximum of comonotone functions and the addition of constants. This functor is a subfunctor of the functor of order-preserving functionals and includes the idempotent measure functor as a subfunctor. The main aim of this paper is to demonstrate that this functor is isomorphic to the capacity functor. We establish this isomorphism using the fuzzy max-plus integral. In essence, this result can be viewed as an idempotent analogue of the Riesz Theorem, which establishes a correspondence between the set of σ-additive regular Borel measures and the set of positive linear functionals.
{"title":"On the functor of comonotonically maxitive functionals","authors":"Taras Radul","doi":"10.1016/j.topol.2024.109131","DOIUrl":"10.1016/j.topol.2024.109131","url":null,"abstract":"<div><div>We introduce a functor of functionals that preserve the maximum of comonotone functions and the addition of constants. This functor is a subfunctor of the functor of order-preserving functionals and includes the idempotent measure functor as a subfunctor. The main aim of this paper is to demonstrate that this functor is isomorphic to the capacity functor. We establish this isomorphism using the fuzzy max-plus integral. In essence, this result can be viewed as an idempotent analogue of the Riesz Theorem, which establishes a correspondence between the set of <em>σ</em>-additive regular Borel measures and the set of positive linear functionals.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"358 ","pages":"Article 109131"},"PeriodicalIF":0.6,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142592851","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}
Pub Date : 2024-11-05DOI: 10.1016/j.topol.2024.109130
Nikola Bogdanovic
In recent years, Barbieri, Dikranjan, Giordano Bruno and Weber have made progress on the problem of determining which characterized subgroups of the circle group are (a-)factorizable, that is, can be written as the sum of two proper (a-)characterized subgroups. We correct an imprecision in one of their results, [2, Theorem 5.9] from 2017, determining the countable a-characterized subgroups of which are also a-factorizable. We also provide a revised proof of [11, Proposition 1.3] (Dikranjan, Kunen, 2007), asserting that is characterized.
近年来，Barbieri、Dikranjan、Giordano Bruno 和 Weber 在确定圆组的哪些特征子群可（a-）因式分解，即可以写成两个适当的（a-）特征子群之和的问题上取得了进展。我们纠正了他们 2017 年的一个结果[2，定理 5.9]中的不精确之处，即确定 T 的可数 a 特征化子群也是可 a 因子化的。我们还对[11，命题 1.3]（Dikranjan，Kunen，2007）进行了修订证明，断言 Q/Z 是可表征的。
{"title":"Some remarks on (a)-characterized subgroups of the circle","authors":"Nikola Bogdanovic","doi":"10.1016/j.topol.2024.109130","DOIUrl":"10.1016/j.topol.2024.109130","url":null,"abstract":"<div><div>In recent years, Barbieri, Dikranjan, Giordano Bruno and Weber have made progress on the problem of determining which characterized subgroups of the circle group are <em>(a-)factorizable</em>, that is, can be written as the sum of two proper (<em>a</em>-)characterized subgroups. We correct an imprecision in one of their results, <span><span>[2, Theorem 5.9]</span></span> from 2017, determining the countable <em>a</em>-characterized subgroups of <span><math><mi>T</mi></math></span> which are also <em>a</em>-factorizable. We also provide a revised proof of <span><span>[11, Proposition 1.3]</span></span> (Dikranjan, Kunen, 2007), asserting that <span><math><mi>Q</mi><mo>/</mo><mi>Z</mi></math></span> is characterized.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"359 ","pages":"Article 109130"},"PeriodicalIF":0.6,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142654963","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}
Pub Date : 2024-10-31DOI: 10.1016/j.topol.2024.109127
Jorge Picado , Aleš Pultr
Open and related maps in the point-free context are studied from a consequently geometric perspective: that is, the opens are concrete well-defined subsets, images of localic maps are set-theoretic images , etc. We present a short proof of Joyal-Tierney Theorem in this setting, a (geometric) characteristic of localic maps that are just complete, and prove that open localic maps also preserve a natural type of sublocales more general than the open ones. A crucial role is played by Frobenius identities that are briefly discussed also in their general aspects.
{"title":"Frobenius identities and geometrical aspects of Joyal-Tierney Theorem","authors":"Jorge Picado , Aleš Pultr","doi":"10.1016/j.topol.2024.109127","DOIUrl":"10.1016/j.topol.2024.109127","url":null,"abstract":"<div><div>Open and related maps in the point-free context are studied from a consequently geometric perspective: that is, the opens are concrete well-defined subsets, images of localic maps are set-theoretic images <span><math><mi>f</mi><mo>[</mo><mi>U</mi><mo>]</mo></math></span>, etc. We present a short proof of Joyal-Tierney Theorem in this setting, a (geometric) characteristic of localic maps that are just complete, and prove that open localic maps also preserve a natural type of sublocales more general than the open ones. A crucial role is played by Frobenius identities that are briefly discussed also in their general aspects.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"358 ","pages":"Article 109127"},"PeriodicalIF":0.6,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142592853","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-30DOI: 10.1016/j.topol.2024.109126
Shimin Li , Jaume Llibre , Qian Tong
Poincaré compactification is very important to investigate the dynamics of vector fields in the neighborhood of the infinity, which is the main concern on the escape of particles to infinity in celestial mechanics, astrophysics, astronomy and some branches of chemistry. Since then Poincaré compactification has been extended into various cases, such as: n-dimensional polynomial vector fields, Hamiltonian vector fields, quasi-homogeneous vector fields, rational vector fields, etc.
In recent years, the piecewise smooth vector fields describing situations with discontinuities such as switching, decisions, impacts etc., have been attracted more and more attention. It is worth to notice that Poincaré compactification has been extended successfully to piecewise polynomial vector fields in 2-dimensional and 3-dimensional cases, and there are also works on n-dimensional Lipschitz continuous vector fields. The main goal of present paper is to extend the Poincaré compactification to n-dimensional piecewise polynomial vector fields which are usually discontinuous, this is a missing point in the existent literature. Thus we can investigate the dynamics near the infinity of n-dimensional piecewise polynomial vector fields. As an application we study the global phase portraits for a class of 3-dimensional piecewise linear differential systems.
庞加莱致密化对于研究无穷邻域矢量场的动力学非常重要，这也是天体力学、天体物理学、天文学和某些化学分支对粒子逸出无穷的主要关注点。从那时起，Poincaré 压缩被扩展到各种情况，如：n 维多项式矢量场、哈密顿矢量场、准均质矢量场、有理矢量场等。近年来，描述具有不连续性的情况（如切换、决策、冲击等）的片状光滑矢量场越来越受到关注。值得注意的是，Poincaré compactification 已成功地扩展到 2 维和 3 维的片断多项式向量场，也有关于 n 维 Lipschitz 连续向量场的研究。本文的主要目标是将普恩卡雷致密化扩展到 n 维的片断多项式矢量场，因为这些矢量场通常是不连续的，而这正是现有文献中缺少的一点。因此，我们可以研究 n 维片断多项式矢量场在无穷大附近的动力学。作为应用，我们研究了一类三维片断线性微分系统的全局相位肖像。
{"title":"Poincaré compactification for n-dimensional piecewise polynomial vector fields: Theory and applications","authors":"Shimin Li , Jaume Llibre , Qian Tong","doi":"10.1016/j.topol.2024.109126","DOIUrl":"10.1016/j.topol.2024.109126","url":null,"abstract":"<div><div>Poincaré compactification is very important to investigate the dynamics of vector fields in the neighborhood of the infinity, which is the main concern on the escape of particles to infinity in celestial mechanics, astrophysics, astronomy and some branches of chemistry. Since then Poincaré compactification has been extended into various cases, such as: <em>n</em>-dimensional polynomial vector fields, Hamiltonian vector fields, quasi-homogeneous vector fields, rational vector fields, etc.</div><div>In recent years, the piecewise smooth vector fields describing situations with discontinuities such as switching, decisions, impacts etc., have been attracted more and more attention. It is worth to notice that Poincaré compactification has been extended successfully to piecewise polynomial vector fields in 2-dimensional and 3-dimensional cases, and there are also works on <em>n</em>-dimensional Lipschitz continuous vector fields. The main goal of present paper is to extend the Poincaré compactification to <em>n</em>-dimensional piecewise polynomial vector fields which are usually discontinuous, this is a missing point in the existent literature. Thus we can investigate the dynamics near the infinity of <em>n</em>-dimensional piecewise polynomial vector fields. As an application we study the global phase portraits for a class of 3-dimensional piecewise linear differential systems.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"358 ","pages":"Article 109126"},"PeriodicalIF":0.6,"publicationDate":"2024-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142586192","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}
Pub Date : 2024-10-29DOI: 10.1016/j.topol.2024.109125
Sebastián Barría
Let and denote the double arrow of Alexandroff and the Sorgenfrey line, respectively. We show that for any , the space of all unions of at most n closed intervals of is not homogeneous. We also prove that the spaces of non-trivial convergent sequences of and are homogeneous. This partially solves an open question of A. Arhangel'skiǐ [1]. In contrast, we show that the space of closed intervals of is homogeneous.
让 A 和 S 分别表示亚历山大罗夫双箭线和索根弗雷线。我们证明，对于任意 n≥1，A 的最多 n 个封闭区间的所有联合的空间不是同质的。我们还证明了 A 和 S 的非琐收敛序列的空间是同质的。这部分解决了 A. Arhangel'skiǐ [1] 的一个未决问题。相反，我们证明了 S 的闭区间空间是同质的。
{"title":"Hyperspaces of the double arrow","authors":"Sebastián Barría","doi":"10.1016/j.topol.2024.109125","DOIUrl":"10.1016/j.topol.2024.109125","url":null,"abstract":"<div><div>Let <span><math><mi>A</mi></math></span> and <span><math><mi>S</mi></math></span> denote the double arrow of Alexandroff and the Sorgenfrey line, respectively. We show that for any <span><math><mi>n</mi><mo>≥</mo><mn>1</mn></math></span>, the space of all unions of at most <em>n</em> closed intervals of <span><math><mi>A</mi></math></span> is not homogeneous. We also prove that the spaces of non-trivial convergent sequences of <span><math><mi>A</mi></math></span> and <span><math><mi>S</mi></math></span> are homogeneous. This partially solves an open question of A. Arhangel'skiǐ <span><span>[1]</span></span>. In contrast, we show that the space of closed intervals of <span><math><mi>S</mi></math></span> is homogeneous.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"358 ","pages":"Article 109125"},"PeriodicalIF":0.6,"publicationDate":"2024-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142592850","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}
Pub Date : 2024-10-29DOI: 10.1016/j.topol.2024.109124
Roya Makrooni , Neda Abbasi
In this paper, we define some qualitative properties of non-autonomous discrete dynamical systems such as orbit shift continuum-wise expansivity, orbit shift persistence and orbit shift α-persistence. Then we discuss the relation between these notions and give necessary examples. Moreover, we prove that every continuum-wise expansive non-autonomous discrete system on a compact metric space is orbit shift continuum-wise expansive.
{"title":"A note on non-autonomous discrete dynamical systems","authors":"Roya Makrooni , Neda Abbasi","doi":"10.1016/j.topol.2024.109124","DOIUrl":"10.1016/j.topol.2024.109124","url":null,"abstract":"<div><div>In this paper, we define some qualitative properties of non-autonomous discrete dynamical systems such as orbit shift continuum-wise expansivity, orbit shift persistence and orbit shift <em>α</em>-persistence. Then we discuss the relation between these notions and give necessary examples. Moreover, we prove that every continuum-wise expansive non-autonomous discrete system on a compact metric space is orbit shift continuum-wise expansive.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"358 ","pages":"Article 109124"},"PeriodicalIF":0.6,"publicationDate":"2024-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142592852","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}
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