{"title":"Closed copies of N in Rω1","authors":"Alan Dow , Klaas Pieter Hart , Jan van Mill , 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}
Pub Date : 2026-02-15Epub Date: 2025-07-07DOI: 10.1016/j.topol.2025.109513
Aleksander Błaszczyk
In this paper there are investigated spaces , where is a family of filters on a set X. In particular there is shown that for countable X the space always has a one-to-one continuous map (=function) onto , but if consists of ultrafilters, then never admits irreducible one-to-one continuous map onto . Moreover, if consists of nowhere dense ultrafilters, then the space does not admit semi-open continuous maps onto . However, if each ultrafilter in is not nowhere dense, then the space possesses a semi-open continuous map onto .
{"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}
Pub Date : 2026-02-15Epub Date: 2025-07-05DOI: 10.1016/j.topol.2025.109500
Justin Tatch Moore
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.
{"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}
Pub Date : 2026-02-15Epub Date: 2025-07-04DOI: 10.1016/j.topol.2025.109507
J. Ka̧kol , A. Leiderman , V.V. Tkachuk
We show that no crowded separable space with the Baire property can be a -space. We give a characterization of the -property of any space X in terms of the function space and construct an example of a pseudocompact -space which does not have the -property. It is also proved that there exists a dense subspace X of the Cantor cube such that is κ-Fréchet–Urysohn while X is not a -space. If CH holds, then there is a separable dense subspace such that is κ-Fréchet–Urysohn but X fails to be a -space.
{"title":"Some applications of the Δ1-property","authors":"J. Ka̧kol , A. Leiderman , 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}
Pub Date : 2026-02-15Epub Date: 2025-07-07DOI: 10.1016/j.topol.2025.109512
S. Garcia-Ferreira , A.H. Tomita , J. Trianon-Fraga
Given a selective ultrafilter , we prove that there exists a p-compact group topology on without nontrivial convergent sequences and a closed subgroup which contains an element not divisible by any ( denotes the direct sum of copies of ). It also shows that does not admit a p-compact topology for any . Given H a group and G a subgroup of which is not n-divisible, for some prime number , but is m-divisible for each prime number , we show that a group topology compatible with cannot be p-compact for any .
{"title":"On p−compact topologies on certain Abelian groups","authors":"S. Garcia-Ferreira , A.H. Tomita , 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>></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}
{"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}
Pub Date : 2026-02-15Epub Date: 2025-07-04DOI: 10.1016/j.topol.2025.109510
Saharon Shelah , Juris Steprāns
It is shown that the existence of a universal function on pairs implies the existence of a universal function for triples.
证明了对上的全称函数的存在意味着三元组上的全称函数的存在。
{"title":"Higher dimensional universal functions from lower dimensional ones","authors":"Saharon Shelah , 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}
Pub Date : 2026-02-01Epub Date: 2025-11-24DOI: 10.1016/j.topol.2025.109672
Khadijeh Ghasemi , Ali S. Janfada , Hadi Zare
Our aim is to study bordism of immersions beyond the metastable range, so . We focus on the case of and show that for any immersion of a nonboundary , M is bordant to and its triple point manifold is a boundary too. By contrast, we show that if is an immersion whose double point manifold is not a boundary then, up to addition in the relevant bordism group, f is bordant to an immersion of a boundary. Along the way, we record that an application of the Gaussian elimination over can be used to determine the submodule of A-annihilated elements in .”. We also review the method of computing the submodule of primitive elements in for X being path connected.
我们的目的是研究在亚稳范围以外的浸入Mn↑R2n−6的谱性，因此n∈{6,7，…，12}。我们着重讨论了n=11的情况，并证明了对于任意浸入的非边界f:M11 - R16， M是与P6×V5的边界，并且它的三点流形也是一个边界。通过对比，我们证明了如果f:M11 - R16是一个浸入，其双点流形不是边界，则在相应的边界群中，f与一个边界的浸入相邻。在此过程中，我们记录到在Z/2上应用高斯消去可以用来确定HnX中a湮灭元素的子模。我们也回顾了在H ^ H ^ QX中，当X是路径连通时，基元子模的计算方法。
{"title":"On the Gaussian elimination and bordism of certain immersions beyond the metastable range","authors":"Khadijeh Ghasemi , Ali S. Janfada , 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}
Pub Date : 2026-02-01Epub Date: 2025-12-09DOI: 10.1016/j.topol.2025.109687
Mykola Lysynskyi, Sergiy Maksymenko
In a recent paper the authors classified differentiable structures on the non-Hausdorff one-dimensional manifold called the line with two origins which is obtained by gluing two copies of the real line via the identity homeomorphism of .
Here we give a classification of differentiable structures on another non-Hausdorff one-dimensional manifold (called letter “Y”) obtained by gluing two copies of via the identity map of positive reals. It turns out that, in contrast to the real line, for every , both manifolds and admit uncountably many pair-wise non-diffeomorphic -structures.
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.
{"title":"Differentiable structures on a union of two open sets","authors":"Mykola Lysynskyi, 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}
Pub Date : 2026-02-01Epub Date: 2025-11-24DOI: 10.1016/j.topol.2025.109668
Faraz Ahmad
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.
{"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}
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