Pub Date : 2024-11-19DOI: 10.1016/j.topol.2024.109144
Valentina Bais
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 -principal bundles over the 4-sphere by means of their Euler and first Pontryagin classes.
{"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}
Pub Date : 2024-11-13DOI: 10.1016/j.topol.2024.109135
J.A. Cruz-Chapital , A.D. Rojas-Sánchez , Á. Tamariz-Mascarúa , H. Villegas-Rodríguez
For a topological space X, let be the cartesian product of copies of the real line with the topology of the uniform convergence on separable subsets of X. In this article we analyze the subspace of of all real-valued continuous functions on X, denoted by . We determine when is dense and when is closed in , and we obtain some results about the Baire property in . Finally, we determine the cellularity of where is the space of ordinal numbers belonging to with its usual order topology.
{"title":"The uniform convergence topology on separable subsets","authors":"J.A. Cruz-Chapital , A.D. Rojas-Sánchez , Á. Tamariz-Mascarúa , 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}
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.2024.109128
H.M.H. Zarenezhad, Javad Jamalzadeh
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.
{"title":"A new entropy on metric spaces with respect to Bourbaki-bounded subsets","authors":"H.M.H. Zarenezhad, 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}
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}
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