Hadi Hassanzada, Hamid Torabi, Hanieh Mirebrahimi, Ameneh Babaee
{"title":"A discrete topological complexity of discrete motion planning","authors":"Hadi Hassanzada, Hamid Torabi, Hanieh Mirebrahimi, Ameneh Babaee","doi":"10.1016/j.topol.2025.109634","DOIUrl":"10.1016/j.topol.2025.109634","url":null,"abstract":"<div><div>In this paper, we present a framework for discrete motion planning tailored for robots that operate in a discrete manner. Furthermore, we extend the concept of <em>r</em>-discrete homotopy as discrete <span><math><mo>(</mo><mi>s</mi><mo>,</mo><mi>r</mi><mo>)</mo></math></span>-homotopy. Utilizing this framework, we investigate the notion of discrete topological complexity, which is defined as the least number of motion planning algorithms necessary for discrete movement. We establish several properties related to discrete topological complexity; for example, we demonstrate that discrete motion planning within a metric space <em>X</em> is feasible if and only if <em>X</em> is a discrete contractible space. Additionally, we show that the discrete topological complexity is solely determined by the strictly discrete homotopy type of the spaces involved.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109634"},"PeriodicalIF":0.5,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145269844","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}
{"title":"On o-free sequences and compacta","authors":"Nathan Carlson","doi":"10.1016/j.topol.2025.109631","DOIUrl":"10.1016/j.topol.2025.109631","url":null,"abstract":"<div><div>We use the notion of an <em>o</em>-free sequence to give new bounds for the cardinality of Hausdorff spaces and regular spaces. There are several implications for compacta. One is that if <em>X</em> is a compactum then <span><math><mi>w</mi><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>o</mi><mi>t</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msup></math></span>, where <span><math><mi>o</mi><mi>t</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is the <em>o</em>-tightness introduced by Tkachenko. Another is 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>o</mi><mi>t</mi><mo>(</mo><mi>X</mi><mo>)</mo><mi>w</mi><msub><mrow><mi>ψ</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msup></math></span> if <em>X</em> is a compactum. This is shown to be a strict improvement of Arhangel'skiĭ's bound <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>ψ</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msup></math></span>. Finally, we show <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>o</mi><mi>t</mi><mo>(</mo><mi>X</mi><mo>)</mo><mi>π</mi><mi>χ</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msup></math></span> if <em>X</em> is a homogeneous compactum. We note <span><math><mi>h</mi><mi>L</mi><msup><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mrow><mi>o</mi><mi>t</mi><mo>(</mo><mi>X</mi><mo>)</mo><mi>π</mi><mi>χ</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msup><mo>≤</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>t</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msup></math></span> for such spaces, where <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>t</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msup></math></span> is de la Vega's bound for the cardinality of homogeneous compacta.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109631"},"PeriodicalIF":0.5,"publicationDate":"2025-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145269842","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}
{"title":"Topologies and fixpoints on weak partial metric spaces","authors":"Mengqiao Huang , Xiaodong Jia , Qingguo Li","doi":"10.1016/j.topol.2025.109601","DOIUrl":"10.1016/j.topol.2025.109601","url":null,"abstract":"<div><div>For a weak partial metric space <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>p</mi><mo>)</mo></math></span>, there is a canonical metric <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> on <em>X</em>, defined as <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>=</mo><mi>max</mi><mo></mo><mo>{</mo><mi>p</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>−</mo><mi>p</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo>)</mo><mo>,</mo><mi>p</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>−</mo><mi>p</mi><mo>(</mo><mi>y</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>}</mo></math></span> for all <span><math><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>X</mi></math></span>. We prove that the partial metric topology and the Scott topology on <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>p</mi><mo>)</mo></math></span> coincide if and only if the metric topology on <span><math><mo>(</mo><mi>X</mi><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></math></span> and the Lawson topology on <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>p</mi><mo>)</mo></math></span> agree, provided that the weak partial metric space <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>p</mi><mo>)</mo></math></span> is a domain in its specialization order and its associated metric space <span><math><mo>(</mo><mi>X</mi><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></math></span> is compact. We also discussed fixpoints of self maps defined on weak partial metric spaces.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109601"},"PeriodicalIF":0.5,"publicationDate":"2025-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145269836","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}
{"title":"A note on crossing changes and delta-moves for virtual knots","authors":"Ryuji Higa","doi":"10.1016/j.topol.2025.109622","DOIUrl":"10.1016/j.topol.2025.109622","url":null,"abstract":"<div><div>We consider the problem of determining whether two given virtual knots can be converted into each other by a sequence of crossing changes or a sequence of Δ-moves. We provide a simple method derived from the <em>r</em>-covering of a virtual knot for approaching this problem. We also give the lower bounds of the Gordian distance for crossing changes and Δ-moves.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109622"},"PeriodicalIF":0.5,"publicationDate":"2025-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145223046","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}
{"title":"Tight contact structures on toroidal plumbed 3-manifolds","authors":"Tanushree Shah, Jonathan Simone","doi":"10.1016/j.topol.2025.109602","DOIUrl":"10.1016/j.topol.2025.109602","url":null,"abstract":"<div><div>We consider tight contact structures on plumbed 3-manifolds with no bad vertices. We discuss how one can count the number of tight contact structures with zero Giroux torsion on such 3-manifolds and explore conditions under which Giroux torsion can be added to these tight contact structures without making them overtwisted. We give an explicit algorithm to construct stein diagrams corresponding to tight structures without Giroux torsion. We focus mainly on plumbed 3-manifolds whose vertices have valence at most 3 and then briefly consider the situation for plumbed 3-manifolds with vertices of higher valence.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109602"},"PeriodicalIF":0.5,"publicationDate":"2025-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145269843","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}
{"title":"Attractors as a bridge from topological properties to long-term behavior in dynamical systems","authors":"Aliasghar Sarizadeh","doi":"10.1016/j.topol.2025.109604","DOIUrl":"10.1016/j.topol.2025.109604","url":null,"abstract":"<div><div>This paper refined and introduced some notations (namely attractors, physical attractors, proper attractors, topologically exact and topologically mixing) within the context of relations. We establish necessary and sufficient conditions, including that the phase space of a topologically exact system is an attractor for its inverse, and vice versa, and that a system is topologically mixing if and only if its phase space is a physical attractor.</div><div>Through iterated function systems (IFSs), we illustrate classes of non-trivial topologically mixing and topologically exact IFSs. Additionally, we use IFSs to provide an example of topologically mixing system, generated by finite of homeomorphisms on a compact metric space, that is not topologically exact. These findings connect topological properties with attractor types, providing deeper insights into the long-term dynamics of such systems.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109604"},"PeriodicalIF":0.5,"publicationDate":"2025-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145223045","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}
{"title":"Rational cohomology and Cartan matrix","authors":"Chi-Heng Zhang , Nan Gao , Zi-Cheng Cheng","doi":"10.1016/j.topol.2025.109603","DOIUrl":"10.1016/j.topol.2025.109603","url":null,"abstract":"<div><div>Gabrel-Krause dimension of the rational cohomology <span><math><msup><mrow><mi>H</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><mi>B</mi><msup><mrow><mi>T</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>;</mo><mi>Q</mi><mo>)</mo></math></span> is described for the <em>m</em>-torus <span><math><msup><mrow><mi>T</mi></mrow><mrow><mi>m</mi></mrow></msup></math></span>. Inspired by the diagonalizability of admissible map between <span><math><msup><mrow><mi>H</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><mi>B</mi><msup><mrow><mi>T</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>,</mo><mi>Q</mi><mo>)</mo></math></span>, the relationship of minimal realization among symmetrizable generalised Cartan matrices is shown.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109603"},"PeriodicalIF":0.5,"publicationDate":"2025-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145269841","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}
{"title":"Some notes on topological rings and their groups of units","authors":"Abolfazl Tarizadeh","doi":"10.1016/j.topol.2025.109600","DOIUrl":"10.1016/j.topol.2025.109600","url":null,"abstract":"<div><div>If <em>R</em> is a topological ring then <span><math><msup><mrow><mi>R</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>, the group of units of <em>R</em>, with the subspace topology is not necessarily a topological group. This leads us to the following natural definition: By absolute topological ring we mean a topological ring such that its group of units with the subspace topology is a topological group. We prove that every commutative ring with the <em>I</em>-adic topology is an absolute topological ring (where <em>I</em> is an ideal of the ring).</div><div>Next, we prove that if <em>I</em> is an ideal of a ring <em>R</em> then for the <em>I</em>-adic topology over <em>R</em> we have <span><math><msub><mrow><mi>π</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo><mo>=</mo><mi>R</mi><mo>/</mo><mo>(</mo><munder><mo>⋂</mo><mrow><mi>n</mi><mo>⩾</mo><mn>1</mn></mrow></munder><msup><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo><mo>=</mo><mi>t</mi><mo>(</mo><mi>R</mi><mo>)</mo></math></span> where <span><math><msub><mrow><mi>π</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math></span> is the space of connected components of <em>R</em> and <span><math><mi>t</mi><mo>(</mo><mi>R</mi><mo>)</mo></math></span> is the space of irreducible closed subsets of <em>R</em>.</div><div>We also show with an example that the identity component of a topological group is not necessarily a characteristic subgroup.</div><div>Finally, we observed that the main result of Koh <span><span>[3]</span></span> as well as its corrected form <span><span>[5, Chap II, §12, Theorem 12.1]</span></span> is not true, and then we corrected this result in the right way.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109600"},"PeriodicalIF":0.5,"publicationDate":"2025-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145183877","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}
{"title":"Extensions of increasing bounded uniformly continuous functions","authors":"Kaori Yamazaki","doi":"10.1016/j.topol.2025.109599","DOIUrl":"10.1016/j.topol.2025.109599","url":null,"abstract":"<div><div>In this paper, we show that, for an increasing bounded uniformly continuous function <em>f</em> on a subspace <em>A</em> of a uniform space <em>X</em> equipped with a preorder, <em>f</em> can be extended to an increasing uniformly continuous function on <em>X</em> if and only if <em>f</em> is uniformly completely order separated in <em>X</em>. This extends McShane's Extension Theorem for metric spaces and Katětov's Theorem for uniform spaces. Moreover, we establish a characterization of a uniform/metric space <em>X</em> equipped with a preorder possessing the monotone uniform extension property, which answers a question asked by E.A.Ok.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109599"},"PeriodicalIF":0.5,"publicationDate":"2025-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145222985","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}
{"title":"Large C*-polyhedrons in C*-algebras","authors":"Clayton Suguio Hida","doi":"10.1016/j.topol.2025.109598","DOIUrl":"10.1016/j.topol.2025.109598","url":null,"abstract":"<div><div>A classical polyhedron in a Banach space is a collection of points with a distinctive geometric separation property: each point in the set can be separated from the others by a closed convex set. This concept reflects the interplay between convexity and the geometry of Banach spaces. In this article, we introduce and study a noncommutative analogue of this notion, based on the concept of <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-convexity, a generalization of classical convexity within the setting of <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-algebras. We define the notion of a <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-polyhedron as a family of elements in a <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-algebra that satisfies a similar separation property with respect to closed <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-convex sets. Our main goal is to investigate the maximal possible cardinality of <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-polyhedrons in various <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-algebras, with particular attention to classical examples from the theory of operator algebras, such as the algebras of compact and bounded operators on a Hilbert space.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"375 ","pages":"Article 109598"},"PeriodicalIF":0.5,"publicationDate":"2025-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145118518","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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