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On Dold-Whitney's parallelizability of 4-manifolds 论多尔德-惠特尼的 4 维平行性
IF 0.6 4区 数学
Topology and its Applications Pub Date : 2024-11-19 DOI: 10.1016/j.topol.2024.109144
Valentina Bais
{"title":"On Dold-Whitney's parallelizability of 4-manifolds","authors":"Valentina Bais","doi":"10.1016/j.topol.2024.109144","DOIUrl":"10.1016/j.topol.2024.109144","url":null,"abstract":"<div><div>We present a proof of a theorem by Dold and Whitney, according to which a closed orientable 4-manifold is parallelizable if and only if its second Stiefel-Whitney class, first Pontryagin class and Euler characteristics vanish. This follows from a stronger result due to Dold and Whitney on the classification of oriented sphere bundles over a 4-complex. Our proof is based on an argument by R. Kirby on the classification of <span><math><mi>S</mi><mi>O</mi><mo>(</mo><mn>4</mn><mo>)</mo></math></span>-principal bundles over the 4-sphere by means of their Euler and first Pontryagin classes.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"359 ","pages":"Article 109144"},"PeriodicalIF":0.6,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142719800","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
The uniform convergence topology on separable subsets 可分离子集上的均匀收敛拓扑学
IF 0.6 4区 数学
Topology and its Applications Pub Date : 2024-11-13 DOI: 10.1016/j.topol.2024.109135
J.A. Cruz-Chapital , A.D. Rojas-Sánchez , Á. Tamariz-Mascarúa , H. Villegas-Rodríguez
{"title":"The uniform convergence topology on separable subsets","authors":"J.A. Cruz-Chapital ,&nbsp;A.D. Rojas-Sánchez ,&nbsp;Á. Tamariz-Mascarúa ,&nbsp;H. Villegas-Rodríguez","doi":"10.1016/j.topol.2024.109135","DOIUrl":"10.1016/j.topol.2024.109135","url":null,"abstract":"<div><div>For a topological space <em>X</em>, let <span><math><msub><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>X</mi></mrow></msup><mo>)</mo></mrow><mrow><mi>s</mi></mrow></msub><mo>:</mo><mo>=</mo><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>X</mi></mrow></msup><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>)</mo></math></span> be the cartesian product of <span><math><mo>|</mo><mi>X</mi><mo>|</mo></math></span> copies of the real line <span><math><mi>R</mi></math></span> with the topology of the uniform convergence on separable subsets of <em>X</em>. In this article we analyze the subspace <span><math><mi>C</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> of <span><math><msub><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>X</mi></mrow></msup><mo>)</mo></mrow><mrow><mi>s</mi></mrow></msub></math></span> of all real-valued continuous functions on <em>X</em>, denoted by <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span>. We determine when <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is dense and when is closed in <span><math><msub><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>X</mi></mrow></msup><mo>)</mo></mrow><mrow><mi>s</mi></mrow></msub></math></span>, and we obtain some results about the Baire property in <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span>. Finally, we determine the cellularity of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>(</mo><mo>[</mo><mn>0</mn><mo>,</mo><mi>α</mi><mo>]</mo><mo>)</mo></math></span> where <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mi>α</mi><mo>]</mo></math></span> is the space of ordinal numbers belonging to <span><math><mi>α</mi><mo>+</mo><mn>1</mn></math></span> with its usual order topology.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"359 ","pages":"Article 109135"},"PeriodicalIF":0.6,"publicationDate":"2024-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142706390","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
Relatively functionally countable subsets of products 产品的相对功能可数子集
IF 0.6 4区 数学
Topology and its Applications Pub Date : 2024-11-12 DOI: 10.1016/j.topol.2024.109133
Anton E. Lipin
{"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}
引用次数: 0
Extendability to Marczewski-Burstin countably representable ideals 扩展到马茨维斯基-布尔斯坦可数可表示理想的可扩展性
IF 0.6 4区 数学
Topology and its Applications Pub Date : 2024-11-09 DOI: 10.1016/j.topol.2024.109134
Marta Kwela, Jacek Tryba
{"title":"Extendability to Marczewski-Burstin countably representable ideals","authors":"Marta Kwela,&nbsp;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}
引用次数: 0
MSNR spaces revisited 重温 MSNR 空间
IF 0.6 4区 数学
Topology and its Applications Pub Date : 2024-11-08 DOI: 10.1016/j.topol.2024.109132
John E. Porter
{"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}
引用次数: 0
On Ψω-factorizable groups 关于Ψω可因子群
IF 0.6 4区 数学
Topology and its Applications Pub Date : 2024-11-06 DOI: 10.1016/j.topol.2024.109129
Heng Zhang , Wenfei Xi , Yaoqiang Wu , Hongling Li
{"title":"On Ψω-factorizable groups","authors":"Heng Zhang ,&nbsp;Wenfei Xi ,&nbsp;Yaoqiang Wu ,&nbsp;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}
引用次数: 0
On the functor of comonotonically maxitive functionals 论最大单调函数的函子
IF 0.6 4区 数学
Topology and its Applications Pub Date : 2024-11-05 DOI: 10.1016/j.topol.2024.109131
Taras Radul
{"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}
引用次数: 0
Some remarks on (a)-characterized subgroups of the circle 关于圆的 (a) 特征子群的一些评论
IF 0.6 4区 数学
Topology and its Applications Pub Date : 2024-11-05 DOI: 10.1016/j.topol.2024.109130
Nikola Bogdanovic
{"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}
引用次数: 0
A new entropy on metric spaces with respect to Bourbaki-bounded subsets 关于bourbaki有界子集的度量空间上的新熵
IF 0.6 4区 数学
Topology and its Applications Pub Date : 2024-10-31 DOI: 10.1016/j.topol.2024.109128
H.M.H. Zarenezhad, Javad Jamalzadeh
{"title":"A new entropy on metric spaces with respect to Bourbaki-bounded subsets","authors":"H.M.H. Zarenezhad,&nbsp;Javad Jamalzadeh","doi":"10.1016/j.topol.2024.109128","DOIUrl":"10.1016/j.topol.2024.109128","url":null,"abstract":"<div><div>In this paper, we define a new entropy for every self-map on metric spaces, which is referred to as the Bourbaki entropy. We show, by means of an example, that the metric entropy is not necessarily equal to the Bourbaki entropy. Finally, the basic properties of the Bourbaki entropy are studied. The obtained results include the logarithmic law, invariance under conjugation, the weak addition theorem, and the completion theorem.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"359 ","pages":"Article 109128"},"PeriodicalIF":0.6,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142743385","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
Frobenius identities and geometrical aspects of Joyal-Tierney Theorem 弗罗贝纽斯等式和乔亚尔-蒂尔尼定理的几何方面
IF 0.6 4区 数学
Topology and its Applications Pub Date : 2024-10-31 DOI: 10.1016/j.topol.2024.109127
Jorge Picado , Aleš Pultr
{"title":"Frobenius identities and geometrical aspects of Joyal-Tierney Theorem","authors":"Jorge Picado ,&nbsp;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}
引用次数: 0
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