{"title":"Weak approximation by points in function spaces and in the power of Arens' space","authors":"Kenichi Tamano , Stevo Todorčević","doi":"10.1016/j.topol.2025.109629","DOIUrl":"10.1016/j.topol.2025.109629","url":null,"abstract":"<div><div>We study the weak approximation by points (WAP) in function spaces <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> and in the power <span><math><msubsup><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>ω</mi></mrow></msubsup></math></span> of Arens' space <span><math><msub><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>. The following two results are shown:</div><div>(1) The space <span><math><msubsup><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>ω</mi></mrow></msubsup></math></span>, which can be embedded in <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><msup><mrow><mi>ω</mi></mrow><mrow><mi>ω</mi></mrow></msup><mo>)</mo></math></span> and <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><msup><mrow><mi>ω</mi></mrow><mrow><mi>ω</mi></mrow></msup><mo>)</mo></math></span>, is WAP, answering a question of G. Gruenhage, B. Tsaban, and L. Zdomskyy.</div><div>(2) <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><msup><mrow><mi>ω</mi></mrow><mrow><mi>ω</mi></mrow></msup><mo>)</mo></math></span> is not WAP.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109629"},"PeriodicalIF":0.5,"publicationDate":"2025-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145321826","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 : 2025-10-03DOI: 10.1016/j.topol.2025.109630
Antoni Machowski
We examine subgroups of locally compact groups that are continuous homomorphic images of connected Lie groups and we give a criterion for being such an image. We also provide a new characterisation of Lie groups and a characterisation of groups that are images of connected locally compact groups.
{"title":"Virtual Lie subgroups of locally compact groups","authors":"Antoni Machowski","doi":"10.1016/j.topol.2025.109630","DOIUrl":"10.1016/j.topol.2025.109630","url":null,"abstract":"<div><div>We examine subgroups of locally compact groups that are continuous homomorphic images of connected Lie groups and we give a criterion for being such an image. We also provide a new characterisation of Lie groups and a characterisation of groups that are images of connected locally compact groups.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109630"},"PeriodicalIF":0.5,"publicationDate":"2025-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145322260","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 : 2025-10-02DOI: 10.1016/j.topol.2025.109601
Mengqiao Huang , Xiaodong Jia , Qingguo Li
For a weak partial metric space , there is a canonical metric on X, defined as for all . We prove that the partial metric topology and the Scott topology on coincide if and only if the metric topology on and the Lawson topology on agree, provided that the weak partial metric space is a domain in its specialization order and its associated metric space is compact. We also discussed fixpoints of self maps defined on weak partial metric spaces.
{"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}
Pub Date : 2025-10-02DOI: 10.1016/j.topol.2025.109622
Ryuji Higa
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 r-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.
{"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}
Pub Date : 2025-10-02DOI: 10.1016/j.topol.2025.109602
Tanushree Shah, Jonathan Simone
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.
{"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}
Pub Date : 2025-10-02DOI: 10.1016/j.topol.2025.109604
Aliasghar Sarizadeh
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.
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.
{"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}
Pub Date : 2025-10-02DOI: 10.1016/j.topol.2025.109628
Nikica Uglešić
A generalization of an inverse system in a category was recently introduced, as well as that of the corresponding pro-category These so called the delay-inverse systems and delay-pro-category could potentially yield a new theory of (delay-) inverse systems as well as and, consequently, a kind of coarser abstract shape theory. However, we have proven that the potential new theory reduces, in its essence (the classification and invariants), to the ordinary one.
{"title":"Note on delay-inverse systems","authors":"Nikica Uglešić","doi":"10.1016/j.topol.2025.109628","DOIUrl":"10.1016/j.topol.2025.109628","url":null,"abstract":"<div><div>A generalization of an inverse system in a category was recently introduced, as well as that of the corresponding pro-category These so called the delay-inverse systems and delay-pro-category could potentially yield a new theory of (delay-) inverse systems as well as and, consequently, a kind of coarser abstract shape theory. However, we have proven that the potential new theory reduces, in its essence (the classification and invariants), to the ordinary one.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109628"},"PeriodicalIF":0.5,"publicationDate":"2025-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145418010","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 : 2025-09-30DOI: 10.1016/j.topol.2025.109603
Chi-Heng Zhang , Nan Gao , Zi-Cheng Cheng
Gabrel-Krause dimension of the rational cohomology is described for the m-torus . Inspired by the diagonalizability of admissible map between , the relationship of minimal realization among symmetrizable generalised Cartan matrices is shown.
{"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}
Pub Date : 2025-09-25DOI: 10.1016/j.topol.2025.109600
Abolfazl Tarizadeh
If R is a topological ring then , the group of units of R, 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 I-adic topology is an absolute topological ring (where I is an ideal of the ring).
Next, we prove that if I is an ideal of a ring R then for the I-adic topology over R we have where is the space of connected components of R and is the space of irreducible closed subsets of R.
We also show with an example that the identity component of a topological group is not necessarily a characteristic subgroup.
Finally, we observed that the main result of Koh [3] as well as its corrected form [5, Chap II, §12, Theorem 12.1] is not true, and then we corrected this result in the right way.
{"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}
Pub Date : 2025-09-24DOI: 10.1016/j.topol.2025.109599
Kaori Yamazaki
In this paper, we show that, for an increasing bounded uniformly continuous function f on a subspace A of a uniform space X equipped with a preorder, f can be extended to an increasing uniformly continuous function on X if and only if f is uniformly completely order separated in X. 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 X equipped with a preorder possessing the monotone uniform extension property, which answers a question asked by E.A.Ok.
{"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}
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号