Pub Date : 2025-12-09DOI: 10.1016/j.topol.2025.109686
Jorge Cruz
Given two infinite cardinals κ and λ, we introduce and study the notion of a κ-barely independent family over λ. We provide some conditions under which these types of families exist. In particular, we relate the existence of large κ-barely independent families with the generalized reaping numbers and use these relations to give conditions under which every uniform ultrafilter over a given cardinal λ is both Tukey top and has maximal character. Finally, we show that implies the non-existence of barely independent families over .
{"title":"κ-Barely independent families and Tukey types of ultrafilters","authors":"Jorge Cruz","doi":"10.1016/j.topol.2025.109686","DOIUrl":"10.1016/j.topol.2025.109686","url":null,"abstract":"<div><div>Given two infinite cardinals <em>κ</em> and <em>λ</em>, we introduce and study the notion of a <em>κ</em>-barely independent family over <em>λ</em>. We provide some conditions under which these types of families exist. In particular, we relate the existence of large <em>κ</em>-barely independent families with the generalized reaping numbers <span><math><mi>r</mi><mo>(</mo><mi>κ</mi><mo>,</mo><mi>λ</mi><mo>)</mo></math></span> and use these relations to give conditions under which every uniform ultrafilter over a given cardinal <em>λ</em> is both Tukey top and has maximal character. Finally, we show that <span><math><mi>p</mi><mo>></mo><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> implies the non-existence of barely independent families over <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"380 ","pages":"Article 109686"},"PeriodicalIF":0.5,"publicationDate":"2025-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145760791","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 : 2025-12-08DOI: 10.1016/j.topol.2025.109685
Koki Iwakura
In this paper, we study the non-singular extension problem of horizontal stable fold maps. This problem asks what conditions ensure the existence of a submersion whose restriction to the boundary coincides with a given map, called a non-singular extension. By defining a combinatorial object called a pairing map, we prove that the existence of a non-singular extension is equivalent to the existence of a pairing map. Furthermore, to facilitate the application of the main theorem, we compute the Euler characteristics and the fundamental groups of compact 3-dimensional manifolds that serve as the source manifolds of non-singular extensions.
{"title":"Non-singular extensions of horizontal stable fold maps from surfaces to the plane","authors":"Koki Iwakura","doi":"10.1016/j.topol.2025.109685","DOIUrl":"10.1016/j.topol.2025.109685","url":null,"abstract":"<div><div>In this paper, we study the non-singular extension problem of horizontal stable fold maps. This problem asks what conditions ensure the existence of a submersion whose restriction to the boundary coincides with a given map, called a non-singular extension. By defining a combinatorial object called a pairing map, we prove that the existence of a non-singular extension is equivalent to the existence of a pairing map. Furthermore, to facilitate the application of the main theorem, we compute the Euler characteristics and the fundamental groups of compact 3-dimensional manifolds that serve as the source manifolds of non-singular extensions.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"378 ","pages":"Article 109685"},"PeriodicalIF":0.5,"publicationDate":"2025-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145737146","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 : 2025-12-08DOI: 10.1016/j.topol.2025.109689
Shahryar Ghaed Sharaf
The main goal of this work is to determine the Betti numbers of the links of isolated singularities in a compact toric variety of real dimension 8, using the CW-structure of the links. Additionally, we construct the intersection spaces associated with these links. Using the duality of the Betti numbers of intersection spaces, we conclude that, similar to the case of toric varieties of real dimension 6, the Betti numbers of the links contain only one non-combinatorial invariant parameter. In the final section, we extend our discussion to arbitrary compact toric varieties and their associated link bundles. We show that for any given link , there exists a fiber bundle with fiber , where the base space X is a compact toric variety. Furthermore, using the Chern–Spanier exact sequences for sphere bundles, we show that for the fiber bundle , where , the non-combinatorial invariant parameters appearing in the Betti numbers of and X are equal. In addition, we provide an algebraic description of the non-combinatorial invariant parameter of X in terms of the cohomological Euler class of the fiber bundle.
{"title":"Link bundles of compact toric varieties of real dimension 8","authors":"Shahryar Ghaed Sharaf","doi":"10.1016/j.topol.2025.109689","DOIUrl":"10.1016/j.topol.2025.109689","url":null,"abstract":"<div><div>The main goal of this work is to determine the Betti numbers of the links of isolated singularities in a compact toric variety of real dimension 8, using the CW-structure of the links. Additionally, we construct the intersection spaces associated with these links. Using the duality of the Betti numbers of intersection spaces, we conclude that, similar to the case of toric varieties of real dimension 6, the Betti numbers of the links contain only one non-combinatorial invariant parameter. In the final section, we extend our discussion to arbitrary compact toric varieties and their associated link bundles. We show that for any given link <span><math><mi>L</mi></math></span>, there exists a fiber bundle <span><math><mi>π</mi><mo>:</mo><mi>L</mi><mo>→</mo><mi>X</mi></math></span> with fiber <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>, where the base space <em>X</em> is a compact toric variety. Furthermore, using the Chern–Spanier exact sequences for sphere bundles, we show that for the fiber bundle <span><math><mi>π</mi><mo>:</mo><mi>L</mi><mo>⟶</mo><mi>X</mi></math></span>, where <span><math><msub><mrow><mi>dim</mi></mrow><mrow><mi>R</mi></mrow></msub><mo></mo><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><mn>6</mn></math></span>, the non-combinatorial invariant parameters appearing in the Betti numbers of <span><math><mi>L</mi></math></span> and <em>X</em> are equal. In addition, we provide an algebraic description of the non-combinatorial invariant parameter of <em>X</em> in terms of the cohomological Euler class of the fiber bundle.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"378 ","pages":"Article 109689"},"PeriodicalIF":0.5,"publicationDate":"2025-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145737147","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 : 2025-12-05DOI: 10.1016/j.topol.2025.109684
Jialiang He, Lili Shen, Yi Zhou
It is shown that the isomorphism relation between continuous t-norms is Borel bireducible with the relation of order isomorphism between linear orders on the set of natural numbers, and therefore, it is a Borel complete equivalence relation.
{"title":"The complexity of classifying continuous t-norms up to isomorphism","authors":"Jialiang He, Lili Shen, Yi Zhou","doi":"10.1016/j.topol.2025.109684","DOIUrl":"10.1016/j.topol.2025.109684","url":null,"abstract":"<div><div>It is shown that the isomorphism relation between continuous t-norms is Borel bireducible with the relation of order isomorphism between linear orders on the set of natural numbers, and therefore, it is a Borel complete equivalence relation.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"378 ","pages":"Article 109684"},"PeriodicalIF":0.5,"publicationDate":"2025-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145737145","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 : 2025-12-04DOI: 10.1016/j.topol.2025.109675
Sourav Bhattacharya, Ashish Yadav
We prove a sufficient condition for a pattern π on a triod Y to have rotation number coincide with an end-point of its forced rotation interval . Then, we demonstrate the existence of peculiar patterns on triods that are neither triod twists nor possess a block structure over a triod twist pattern, but their rotation numbers are an end point of their respective forced rotation intervals, mimicking the behavior of triod twist patterns. These patterns, absent in circle maps (see [1]), highlight a key difference between the rotation theories for triods (introduced in [10]) and that of circle maps. We name these patterns: “strangely ordered” and show that they are semi-conjugate to circle rotations via a piece-wise monotone map. We conclude by providing an algorithm to construct unimodal strangely ordered patterns with arbitrary rotation pairs.
{"title":"Twist like behavior in non-twist patterns of triods","authors":"Sourav Bhattacharya, Ashish Yadav","doi":"10.1016/j.topol.2025.109675","DOIUrl":"10.1016/j.topol.2025.109675","url":null,"abstract":"<div><div>We prove a sufficient condition for a <em>pattern π</em> on a <em>triod Y</em> to have <em>rotation number</em> <span><math><msub><mrow><mi>ρ</mi></mrow><mrow><mi>π</mi></mrow></msub></math></span> coincide with an end-point of its <em>forced rotation interval</em> <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>π</mi></mrow></msub></math></span>. Then, we demonstrate the existence of peculiar <em>patterns</em> on <em>triods</em> that are neither <em>triod twists</em> nor possess a <em>block structure</em> over a <em>triod twist pattern</em>, but their <em>rotation numbers</em> are an end point of their respective <em>forced rotation intervals</em>, mimicking the behavior of <em>triod twist patterns</em>. These <em>patterns</em>, absent in circle maps (see <span><span>[1]</span></span>), highlight a key difference between the rotation theories for <em>triods</em> (introduced in <span><span>[10]</span></span>) and that of circle maps. We name these <em>patterns</em>: “<em>strangely ordered</em>” and show that they are semi-conjugate to circle rotations via a piece-wise monotone map. We conclude by providing an algorithm to construct unimodal <em>strangely ordered patterns</em> with arbitrary <em>rotation pairs</em>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"378 ","pages":"Article 109675"},"PeriodicalIF":0.5,"publicationDate":"2025-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145737141","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}
In this paper we introduce relative versions of several compact-like properties and study their relations and their behavior under the standard topological operations. We also study the preservation of such relative properties under the generation of hyperspaces. Particularly, we give examples to prove that ω-hyperboundedness is not preserved under continuous functions and pseudo-ω-boundedness is not inherited by dense subspaces. Besides, for a normal space X, we prove the following results for its hyperspace of closed sets : if X is p-pseudocompact, then is strongly p-pseudocompact; and, if X is ultrapseudocompact, then is pseudo-ω-bounded.
{"title":"Compact-like properties, their relative versions and hyperspaces","authors":"Irvin Enrique Soberano-González , Gerardo Delgadillo-Piñón , Yasser Fermán Ortíz-Castillo , Reynaldo Rojas-Hernández","doi":"10.1016/j.topol.2025.109674","DOIUrl":"10.1016/j.topol.2025.109674","url":null,"abstract":"<div><div>In this paper we introduce relative versions of several compact-like properties and study their relations and their behavior under the standard topological operations. We also study the preservation of such relative properties under the generation of hyperspaces. Particularly, we give examples to prove that <em>ω</em>-hyperboundedness is not preserved under continuous functions and pseudo-<em>ω</em>-boundedness is not inherited by dense subspaces. Besides, for a normal space <em>X</em>, we prove the following results for its hyperspace of closed sets <span><math><mrow><mi>CL</mi></mrow><mo>(</mo><mi>X</mi><mo>)</mo></math></span>: if <em>X</em> is <em>p</em>-pseudocompact, then <span><math><mrow><mi>CL</mi></mrow><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is strongly <em>p</em>-pseudocompact; and, if <em>X</em> is ultrapseudocompact, then <span><math><mrow><mi>CL</mi></mrow><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is pseudo-<em>ω</em>-bounded.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"378 ","pages":"Article 109674"},"PeriodicalIF":0.5,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145685270","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 : 2025-11-26DOI: 10.1016/j.topol.2025.109666
Shou Lin , Ying Ge , Xiangeng Zhou
This note provides a corrigendum to the proof of Theorem 4.5 in Topol. Appl. 271 (2020) 107049.
本注释提供了对Topol中定理4.5的证明的更正。应用程序271(2020)107049。
{"title":"Corrigendum to “Compact-star networks and the images of metric spaces under C-mappings” [Topol. Appl. 271 (2020) 107049]","authors":"Shou Lin , Ying Ge , Xiangeng Zhou","doi":"10.1016/j.topol.2025.109666","DOIUrl":"10.1016/j.topol.2025.109666","url":null,"abstract":"<div><div>This note provides a corrigendum to the proof of Theorem 4.5 in Topol. Appl. 271 (2020) 107049.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109666"},"PeriodicalIF":0.5,"publicationDate":"2025-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145617797","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 : 2025-11-26DOI: 10.1016/j.topol.2025.109671
Er-Guang Yang
In this paper, we introduce the notion of (strongly) countably metacompact frames as the generalization of countably paracompact frames. We show that our definition of a countably metacompact frame is conservative. Characterizations of such frames in terms of real functions are also presented.
{"title":"On countable metacompactness in point-free topology","authors":"Er-Guang Yang","doi":"10.1016/j.topol.2025.109671","DOIUrl":"10.1016/j.topol.2025.109671","url":null,"abstract":"<div><div>In this paper, we introduce the notion of (strongly) countably metacompact frames as the generalization of countably paracompact frames. We show that our definition of a countably metacompact frame is conservative. Characterizations of such frames in terms of real functions are also presented.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"378 ","pages":"Article 109671"},"PeriodicalIF":0.5,"publicationDate":"2025-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145618476","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 : 2025-11-26DOI: 10.1016/j.topol.2025.109669
Tomoo Yokoyama
This paper gives a topological characterization of Hamiltonian flows with finitely many singular points on compact surfaces, using the concept of “demi-caractéristique” in the sense of Poincaré. Furthermore, we clarify the relationships and distinctions among the Hamiltonian, divergence-free, and non-wandering properties for continuous flows, which gives an affirmative answer to the problem posed by Nikolaev and Zhuzhoma under the assumption of finitely many singular points.
{"title":"Relations among Hamiltonian, area-preserving, and non-wandering flows on compact surfaces","authors":"Tomoo Yokoyama","doi":"10.1016/j.topol.2025.109669","DOIUrl":"10.1016/j.topol.2025.109669","url":null,"abstract":"<div><div>This paper gives a topological characterization of Hamiltonian flows with finitely many singular points on compact surfaces, using the concept of “demi-caractéristique” in the sense of Poincaré. Furthermore, we clarify the relationships and distinctions among the Hamiltonian, divergence-free, and non-wandering properties for continuous flows, which gives an affirmative answer to the problem posed by Nikolaev and Zhuzhoma under the assumption of finitely many singular points.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"378 ","pages":"Article 109669"},"PeriodicalIF":0.5,"publicationDate":"2025-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145685272","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 : 2025-11-26DOI: 10.1016/j.topol.2025.109673
Yuan Gao, Bin Pang
The primary objective of this paper is to study the properties of ⊤-closed sets and the largest ⊤-compactification of ⊤-convergence spaces. Firstly, we study some properties of ⊤-ultrafilters and introduce the concept of a ⊤-closed set and the concept of a ⊤-compact set, examining the relationship between them. Secondly, we present the notion of essentially ⊤-compact ⊤-convergence spaces and explore the necessary and sufficient conditions for a ⊤-convergence space to have the largest ⊤-compactification. Finally, we construct the Richardson ⊤-compactification of a ⊤-convergence space and identify the necessary and sufficient conditions for the Richardson ⊤-compactification to be the largest ⊤-compactification within the framework of Kent ⊤-convergence spaces.
{"title":"The largest T2 ⊤-compactification of ⊤-convergence spaces","authors":"Yuan Gao, Bin Pang","doi":"10.1016/j.topol.2025.109673","DOIUrl":"10.1016/j.topol.2025.109673","url":null,"abstract":"<div><div>The primary objective of this paper is to study the properties of ⊤-closed sets and the largest <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> ⊤-compactification of ⊤-convergence spaces. Firstly, we study some properties of ⊤-ultrafilters and introduce the concept of a ⊤-closed set and the concept of a ⊤-compact set, examining the relationship between them. Secondly, we present the notion of essentially ⊤-compact ⊤-convergence spaces and explore the necessary and sufficient conditions for a ⊤-convergence space to have the largest <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> ⊤-compactification. Finally, we construct the Richardson ⊤-compactification of a ⊤-convergence space and identify the necessary and sufficient conditions for the Richardson ⊤-compactification to be the largest <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> ⊤-compactification within the framework of Kent ⊤-convergence spaces.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"378 ","pages":"Article 109673"},"PeriodicalIF":0.5,"publicationDate":"2025-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145685271","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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