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Closed copies of N in Rω1 Rω1中N的闭合副本
IF 0.5 4区 数学
Topology and its Applications Pub Date : 2026-02-15 Epub Date: 2025-07-07 DOI: 10.1016/j.topol.2025.109514
Alan Dow , Klaas Pieter Hart , Jan van Mill , Hans Vermeer
{"title":"Closed copies of N in Rω1","authors":"Alan Dow ,&nbsp;Klaas Pieter Hart ,&nbsp;Jan van Mill ,&nbsp;Hans Vermeer","doi":"10.1016/j.topol.2025.109514","DOIUrl":"10.1016/j.topol.2025.109514","url":null,"abstract":"<div><div>We investigate closed copies of <span><math><mi>N</mi></math></span> in powers of <span><math><mi>R</mi></math></span> with respect to <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>- and <em>C</em>-embedding. We show that <span><math><msup><mrow><mi>R</mi></mrow><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup></math></span> contains closed copies of <span><math><mi>N</mi></math></span> that are not <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>-embedded.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"379 ","pages":"Article 109514"},"PeriodicalIF":0.5,"publicationDate":"2026-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145948053","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
New applications of the space Seq I 空间序列的新应用
IF 0.5 4区 数学
Topology and its Applications Pub Date : 2026-02-15 Epub Date: 2025-07-07 DOI: 10.1016/j.topol.2025.109513
Aleksander Błaszczyk
{"title":"New applications of the space Seq I","authors":"Aleksander Błaszczyk","doi":"10.1016/j.topol.2025.109513","DOIUrl":"10.1016/j.topol.2025.109513","url":null,"abstract":"<div><div>In this paper there are investigated spaces <span><math><mi>Seq</mi><mspace></mspace><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>U</mi><mo>)</mo></mrow></math></span>, where <span><math><mi>U</mi></math></span> is a family of filters on a set <em>X</em>. In particular there is shown that for countable <em>X</em> the space <span><math><mi>Seq</mi><mspace></mspace><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>U</mi><mo>)</mo></mrow></math></span> always has a one-to-one continuous map (=function) onto <span><math><mi>Q</mi></math></span>, but if <span><math><mi>U</mi></math></span> consists of ultrafilters, then <span><math><mi>Seq</mi><mspace></mspace><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>U</mi><mo>)</mo></mrow></math></span> never admits irreducible one-to-one continuous map onto <span><math><mi>Q</mi></math></span>. Moreover, if <span><math><mi>U</mi></math></span> consists of nowhere dense ultrafilters, then the space <span><math><mi>Seq</mi><mspace></mspace><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>U</mi><mo>)</mo></mrow></math></span> does not admit semi-open continuous maps onto <span><math><mi>Q</mi></math></span>. However, if each ultrafilter in <span><math><mi>U</mi></math></span> is not nowhere dense, then the space <span><math><mi>Seq</mi><mspace></mspace><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>U</mi><mo>)</mo></mrow></math></span> possesses a semi-open continuous map onto <span><math><mi>Q</mi></math></span>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"379 ","pages":"Article 109513"},"PeriodicalIF":0.5,"publicationDate":"2026-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145948064","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 method of forcing 强迫法
IF 0.5 4区 数学
Topology and its Applications Pub Date : 2026-02-15 Epub Date: 2025-07-05 DOI: 10.1016/j.topol.2025.109500
Justin Tatch Moore
{"title":"The method of forcing","authors":"Justin Tatch Moore","doi":"10.1016/j.topol.2025.109500","DOIUrl":"10.1016/j.topol.2025.109500","url":null,"abstract":"<div><div>The purpose of this article is to give a presentation of the method of forcing aimed at someone with little or no knowledge of set theory and logic. The emphasis will be on how the method can be used to prove theorems as opposed to consistency results.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"379 ","pages":"Article 109500"},"PeriodicalIF":0.5,"publicationDate":"2026-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145948056","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 applications of the Δ1-property 一些应用的Δ1-property
IF 0.5 4区 数学
Topology and its Applications Pub Date : 2026-02-15 Epub Date: 2025-07-04 DOI: 10.1016/j.topol.2025.109507
J. Ka̧kol , A. Leiderman , V.V. Tkachuk
{"title":"Some applications of the Δ1-property","authors":"J. Ka̧kol ,&nbsp;A. Leiderman ,&nbsp;V.V. Tkachuk","doi":"10.1016/j.topol.2025.109507","DOIUrl":"10.1016/j.topol.2025.109507","url":null,"abstract":"<div><div>We show that no crowded separable space with the Baire property can be a <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-space. We give a characterization of the <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-property of any space <em>X</em> in terms of the function space <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> and construct an example of a pseudocompact <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-space which does not have the <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-property. It is also proved that there exists a dense subspace <em>X</em> of the Cantor cube <span><math><msup><mrow><mi>D</mi></mrow><mrow><mi>c</mi></mrow></msup></math></span> such that <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is <em>κ</em>-Fréchet–Urysohn while <em>X</em> is not a <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-space. If CH holds, then there is a <em>separable</em> dense subspace <span><math><mi>X</mi><mo>⊂</mo><msup><mrow><mi>D</mi></mrow><mrow><mi>c</mi></mrow></msup></math></span> such that <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is <em>κ</em>-Fréchet–Urysohn but <em>X</em> fails to be a <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-space.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"379 ","pages":"Article 109507"},"PeriodicalIF":0.5,"publicationDate":"2026-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145948062","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 p−compact topologies on certain Abelian groups 某些阿贝尔群上的p -紧拓扑
IF 0.5 4区 数学
Topology and its Applications Pub Date : 2026-02-15 Epub Date: 2025-07-07 DOI: 10.1016/j.topol.2025.109512
S. Garcia-Ferreira , A.H. Tomita , J. Trianon-Fraga
{"title":"On p−compact topologies on certain Abelian groups","authors":"S. Garcia-Ferreira ,&nbsp;A.H. Tomita ,&nbsp;J. Trianon-Fraga","doi":"10.1016/j.topol.2025.109512","DOIUrl":"10.1016/j.topol.2025.109512","url":null,"abstract":"<div><div>Given a selective ultrafilter <span><math><mi>p</mi><mo>∈</mo><msup><mrow><mi>ω</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>, we prove that there exists a <em>p</em>-compact group topology on <span><math><msup><mrow><mi>Q</mi></mrow><mrow><mo>(</mo><mi>c</mi><mo>)</mo></mrow></msup></math></span> without nontrivial convergent sequences and a closed subgroup <span><math><mi>H</mi><mo>⊆</mo><msup><mrow><mi>Q</mi></mrow><mrow><mo>(</mo><mi>c</mi><mo>)</mo></mrow></msup></math></span> which contains an element not divisible by any <span><math><mi>n</mi><mo>∈</mo><mi>ω</mi></math></span> (<span><math><msup><mrow><mi>Q</mi></mrow><mrow><mo>(</mo><mi>c</mi><mo>)</mo></mrow></msup></math></span> denotes the direct sum of <span><math><mi>c</mi></math></span> copies of <span><math><mi>Q</mi></math></span>). It also shows that <span><math><mi>Z</mi><mo>⊕</mo><msup><mrow><mi>Q</mi></mrow><mrow><mo>(</mo><mi>c</mi><mo>)</mo></mrow></msup></math></span> does not admit a <em>p</em>-compact topology for any <span><math><mi>p</mi><mo>∈</mo><msup><mrow><mi>ω</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>. Given <em>H</em> a group and <em>G</em> a subgroup of <span><math><mi>Q</mi></math></span> which is not <em>n</em>-divisible, for some prime number <span><math><mi>n</mi><mo>&gt;</mo><mn>1</mn></math></span>, but is <em>m</em>-divisible for each prime number <span><math><mi>m</mi><mo>≠</mo><mi>n</mi></math></span>, we show that a group topology compatible with <span><math><mi>H</mi><mo>⊕</mo><mi>G</mi></math></span> cannot be <em>p</em>-compact for any <span><math><mi>p</mi><mo>∈</mo><msup><mrow><mi>ω</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"379 ","pages":"Article 109512"},"PeriodicalIF":0.5,"publicationDate":"2026-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145948063","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 non-CH inner approach construction 非ch内部方法构造
IF 0.5 4区 数学
Topology and its Applications Pub Date : 2026-02-15 Epub Date: 2025-07-04 DOI: 10.1016/j.topol.2025.109509
Yinhe Peng
{"title":"A non-CH inner approach construction","authors":"Yinhe Peng","doi":"10.1016/j.topol.2025.109509","DOIUrl":"10.1016/j.topol.2025.109509","url":null,"abstract":"<div><div>We introduce a new way of constructing <em>γ</em>-sets. Among others, we construct a model in which <span><math><mrow><mi>MA</mi></mrow><mo>+</mo><mo>¬</mo></math></span>CH holds and there are <em>γ</em>-sets <span><math><mi>X</mi><mo>,</mo><mi>Y</mi><mo>⊆</mo><mi>R</mi></math></span> such that <span><math><mi>X</mi><mo>×</mo><mi>Y</mi></math></span> is not a <em>γ</em>-set.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"379 ","pages":"Article 109509"},"PeriodicalIF":0.5,"publicationDate":"2026-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145948030","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
Higher dimensional universal functions from lower dimensional ones 高维泛函数从低维泛函数衍生而来
IF 0.5 4区 数学
Topology and its Applications Pub Date : 2026-02-15 Epub Date: 2025-07-04 DOI: 10.1016/j.topol.2025.109510
Saharon Shelah , Juris Steprāns
{"title":"Higher dimensional universal functions from lower dimensional ones","authors":"Saharon Shelah ,&nbsp;Juris Steprāns","doi":"10.1016/j.topol.2025.109510","DOIUrl":"10.1016/j.topol.2025.109510","url":null,"abstract":"<div><div>It is shown that the existence of a universal function on pairs implies the existence of a universal function for triples.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"379 ","pages":"Article 109510"},"PeriodicalIF":0.5,"publicationDate":"2026-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145948031","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 Gaussian elimination and bordism of certain immersions beyond the metastable range 超出亚稳范围的某些浸渍的高斯消去和边界
IF 0.5 4区 数学
Topology and its Applications Pub Date : 2026-02-01 Epub Date: 2025-11-24 DOI: 10.1016/j.topol.2025.109672
Khadijeh Ghasemi , Ali S. Janfada , Hadi Zare
{"title":"On the Gaussian elimination and bordism of certain immersions beyond the metastable range","authors":"Khadijeh Ghasemi ,&nbsp;Ali S. Janfada ,&nbsp;Hadi Zare","doi":"10.1016/j.topol.2025.109672","DOIUrl":"10.1016/j.topol.2025.109672","url":null,"abstract":"<div><div>Our aim is to study bordism of immersions <span><math><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>↬</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>6</mn></mrow></msup></math></span> beyond the metastable range, so <span><math><mi>n</mi><mo>∈</mo><mo>{</mo><mn>6</mn><mo>,</mo><mn>7</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>12</mn><mo>}</mo></math></span>. We focus on the case of <span><math><mi>n</mi><mo>=</mo><mn>11</mn></math></span> and show that for any immersion of a nonboundary <span><math><mi>f</mi><mo>:</mo><msup><mrow><mi>M</mi></mrow><mrow><mn>11</mn></mrow></msup><mo>↬</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>16</mn></mrow></msup></math></span>, <em>M</em> is bordant to <span><math><msup><mrow><mi>P</mi></mrow><mrow><mn>6</mn></mrow></msup><mo>×</mo><msup><mrow><mi>V</mi></mrow><mrow><mn>5</mn></mrow></msup></math></span> and its triple point manifold is a boundary too. By contrast, we show that if <span><math><mi>f</mi><mo>:</mo><msup><mrow><mi>M</mi></mrow><mrow><mn>11</mn></mrow></msup><mo>↬</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>16</mn></mrow></msup></math></span> is an immersion whose double point manifold is not a boundary then, up to addition in the relevant bordism group, <em>f</em> is bordant to an immersion of a boundary. Along the way, we record that an application of the Gaussian elimination over <span><math><mi>Z</mi><mo>/</mo><mn>2</mn></math></span> can be used to determine the submodule of A-annihilated elements in <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub><mi>X</mi></math></span>.”. We also review the method of computing the submodule of primitive elements in <span><math><msub><mrow><mi>H</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mi>Q</mi><mi>X</mi></math></span> for <em>X</em> being path connected.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"378 ","pages":"Article 109672"},"PeriodicalIF":0.5,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145600404","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
Differentiable structures on a union of two open sets 两个开集并上的可微结构
IF 0.5 4区 数学
Topology and its Applications Pub Date : 2026-02-01 Epub Date: 2025-12-09 DOI: 10.1016/j.topol.2025.109687
Mykola Lysynskyi, Sergiy Maksymenko
{"title":"Differentiable structures on a union of two open sets","authors":"Mykola Lysynskyi,&nbsp;Sergiy Maksymenko","doi":"10.1016/j.topol.2025.109687","DOIUrl":"10.1016/j.topol.2025.109687","url":null,"abstract":"<div><div>In a recent paper the authors classified differentiable structures on the non-Hausdorff one-dimensional manifold <span><math><mi>L</mi></math></span> called the <em>line with two origins</em> which is obtained by gluing two copies of the real line <span><math><mi>R</mi></math></span> via the identity homeomorphism of <span><math><mi>R</mi><mo>∖</mo><mn>0</mn></math></span>.</div><div>Here we give a classification of differentiable structures on another non-Hausdorff one-dimensional manifold <span><math><mi>Y</mi></math></span> (called <em>letter</em> “<em>Y</em>”) obtained by gluing two copies of <span><math><mi>R</mi></math></span> via the identity map of positive reals. It turns out that, in contrast to the real line, for every <span><math><mi>r</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mo>∞</mo></math></span>, both manifolds <span><math><mi>L</mi></math></span> and <span><math><mi>Y</mi></math></span> admit uncountably many pair-wise non-diffeomorphic <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msup></math></span>-structures.</div><div>We also observe that the proofs of these classifications are very similar. This allows to formalize the arguments and extend them to a certain general statement about arrows in arbitrary categories.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"378 ","pages":"Article 109687"},"PeriodicalIF":0.5,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145790739","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
Compactification of perception pairs and spaces of group equivariant non-expansive operators 群等变非膨胀算子的感知对与空间的紧化
IF 0.5 4区 数学
Topology and its Applications Pub Date : 2026-02-01 Epub Date: 2025-11-24 DOI: 10.1016/j.topol.2025.109668
Faraz Ahmad
{"title":"Compactification of perception pairs and spaces of group equivariant non-expansive operators","authors":"Faraz Ahmad","doi":"10.1016/j.topol.2025.109668","DOIUrl":"10.1016/j.topol.2025.109668","url":null,"abstract":"<div><div>We study some metric and topological properties of spaces of equivariant operators. We define the notions of a compact perception pair, compactification of a perception pair, and compactification of a space of group equivariant non-expansive operators. We prove that every perception pair with totally bounded space of measurements, which is also rich enough to endow the common domain with a metric structure, can be isometrically embedded in a compact perception pair. Likewise, we prove that if the images of group equivariant non-expansive operators in a given space form a cover for their common codomain, then the space of such operators can be isometrically embedded in a compact space of group equivariant non-expansive operators, such that the new reference perception pairs are compactifications of the original ones having totally bounded data sets. Meanwhile, we state some compatibility conditions for these embeddings and show that they too are satisfied by our constructions.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"378 ","pages":"Article 109668"},"PeriodicalIF":0.5,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145685269","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
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