Pub Date : 2026-01-14DOI: 10.1016/j.topol.2026.109732
Bryce Decker , Nathan Dalaklis
We assign every metric space X the value , an ordinal number or one of the symbols −1 or Ω, and we call it the D-variant of transfinite Hausdorff dimension of X. This ordinal assignment is primarily constructed by way of the D-dimension, a transfinite dimension function consistent with the large inductive dimension on finite dimensional metric spaces while also addressing shortcomings of the large transfinite inductive dimension. Similar to Hausdorff dimension, is monotone with respect to subspaces, and is a bi-Lipschitz invariant. It is also non-increasing with respect to Lipschitz maps and satisfies a coarse intermediate dimension property. We also show that this new transfinite Hausdorff dimension function addresses the primary goal of transfinite Hausdorff dimension functions; to classify metric spaces with infinite Hausdorff dimension. In particular, we show that if , then . for any separable metric space, and that one can find a metrizable space with bounded between a given ordinal and its successive cardinal with topological dimension 0.
{"title":"The D-variant of transfinite Hausdorff dimension","authors":"Bryce Decker , Nathan Dalaklis","doi":"10.1016/j.topol.2026.109732","DOIUrl":"10.1016/j.topol.2026.109732","url":null,"abstract":"<div><div>We assign every metric space <em>X</em> the value <span><math><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>D</mi></mrow></msub><mi>HD</mi></mrow><mo>(</mo><mi>X</mi><mo>)</mo></math></span>, an ordinal number or one of the symbols −1 or Ω, and we call it the <em>D</em>-variant of transfinite Hausdorff dimension of <em>X</em>. This ordinal assignment is primarily constructed by way of the <em>D</em>-dimension, a transfinite dimension function consistent with the large inductive dimension on finite dimensional metric spaces while also addressing shortcomings of the large transfinite inductive dimension. Similar to Hausdorff dimension, <span><math><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>D</mi></mrow></msub><mi>HD</mi></mrow><mo>(</mo><mo>⋅</mo><mo>)</mo></math></span> is monotone with respect to subspaces, and is a bi-Lipschitz invariant. It is also non-increasing with respect to Lipschitz maps and satisfies a coarse intermediate dimension property. We also show that this new transfinite Hausdorff dimension function addresses the primary goal of transfinite Hausdorff dimension functions; to classify metric spaces with infinite Hausdorff dimension. In particular, we show that if <span><math><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>D</mi></mrow></msub><mi>HD</mi></mrow><mo>≥</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>, then <span><math><mi>HD</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><mo>∞</mo></math></span>. <span><math><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>D</mi></mrow></msub><mi>HD</mi></mrow><mo>(</mo><mi>X</mi><mo>)</mo><mo><</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> for any separable metric space, and that one can find a metrizable space with <span><math><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>D</mi></mrow></msub><mi>HD</mi></mrow><mo>(</mo><mi>X</mi><mo>)</mo></math></span> bounded between a given ordinal and its successive cardinal with topological dimension 0.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"380 ","pages":"Article 109732"},"PeriodicalIF":0.5,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146038156","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 : 2026-01-14DOI: 10.1016/j.topol.2026.109711
Sina Greenwood , Michael Lockyer
In this paper we investigate conditions for an inverse limit of set-valued functions on intervals to be a graph, and in particular an arc or a circle. We analyse how ramification points are formed and give a characterisation of the order of a point in an inverse limit of set-valued functions that is a finite graph, and we strengthen a result by Nall and Vidal-Escobar who showed that if an inverse limit of set-valued functions on intervals is a finite graph, then it is homeomorphic to the Mahavier product of the first n functions of the sequence for some . Recently the notion of a splitting sequence was introduced to provide a characterisation of inverse limits on intervals that are arcs. We survey necessary conditions for a set-valued inverse limit to be an arc or circle which includes a generalisation of this notion.
{"title":"Arcs, circles, finite graphs and inverse limits of set-valued functions on intervals","authors":"Sina Greenwood , Michael Lockyer","doi":"10.1016/j.topol.2026.109711","DOIUrl":"10.1016/j.topol.2026.109711","url":null,"abstract":"<div><div>In this paper we investigate conditions for an inverse limit of set-valued functions on intervals to be a graph, and in particular an arc or a circle. We analyse how ramification points are formed and give a characterisation of the order of a point in an inverse limit of set-valued functions that is a finite graph, and we strengthen a result by Nall and Vidal-Escobar who showed that if an inverse limit of set-valued functions on intervals is a finite graph, then it is homeomorphic to the Mahavier product of the first <em>n</em> functions of the sequence for some <span><math><mi>n</mi><mo>∈</mo><mi>N</mi></math></span>. Recently the notion of a splitting sequence was introduced to provide a characterisation of inverse limits on intervals that are arcs. We survey necessary conditions for a set-valued inverse limit to be an arc or circle which includes a generalisation of this notion.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"380 ","pages":"Article 109711"},"PeriodicalIF":0.5,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146038157","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 : 2026-01-14DOI: 10.1016/j.topol.2026.109730
Yazmin Cote, Carlos Uzcátegui-Aylwin
This article explores the connections between graph-theoretical and topological approaches in the study of the Jordan curve theorem for grids. Building on the foundational work of Rosenfeld, who developed adjacency-based concepts on , and the subsequent introduction of the topological digital plane with the Khalimsky topology by Khalimsky, Kopperman, and Meyer, we investigate the interplay between these perspectives. Inspired by the work of Khalimsky, Kopperman, and Meyer, we define an operator transforming subsets of into subsets of . This operator is essential for demonstrating how 8-paths, 4-connectivity, and other discrete structures in correspond to topological properties in . Moreover, we address whether the topological Jordan curve theorem for can be derived from the graph-theoretical version on . Our results illustrate the deep and intricate relationship between these two methodologies, shedding light on their complementary roles in digital topology.
{"title":"Bridging graph-theoretical and topological approaches: Connectivity and Jordan curves in the digital plane","authors":"Yazmin Cote, Carlos Uzcátegui-Aylwin","doi":"10.1016/j.topol.2026.109730","DOIUrl":"10.1016/j.topol.2026.109730","url":null,"abstract":"<div><div>This article explores the connections between graph-theoretical and topological approaches in the study of the Jordan curve theorem for grids. Building on the foundational work of Rosenfeld, who developed adjacency-based concepts on <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, and the subsequent introduction of the topological digital plane <span><math><msup><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> with the Khalimsky topology by Khalimsky, Kopperman, and Meyer, we investigate the interplay between these perspectives. Inspired by the work of Khalimsky, Kopperman, and Meyer, we define an operator <span><math><msup><mrow><mi>Γ</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> transforming subsets of <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> into subsets of <span><math><msup><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. This operator is essential for demonstrating how 8-paths, 4-connectivity, and other discrete structures in <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> correspond to topological properties in <span><math><msup><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. Moreover, we address whether the topological Jordan curve theorem for <span><math><msup><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> can be derived from the graph-theoretical version on <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. Our results illustrate the deep and intricate relationship between these two methodologies, shedding light on their complementary roles in digital topology.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"380 ","pages":"Article 109730"},"PeriodicalIF":0.5,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146038155","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 : 2026-01-13DOI: 10.1016/j.topol.2026.109731
Hugo Juárez-Anguiano , Raúl Juárez-Flores
In this paper, we prove the following result: Let H be a closed subgroup of a compact metrizable group G. Then is G-movable if and only if H is a large subgroup of G. It provides a new characterization of large subgroups and generalizes a result of Gevorgyan [12] about compact Lie groups.
{"title":"G-movability and large subgroups","authors":"Hugo Juárez-Anguiano , Raúl Juárez-Flores","doi":"10.1016/j.topol.2026.109731","DOIUrl":"10.1016/j.topol.2026.109731","url":null,"abstract":"<div><div>In this paper, we prove the following result: Let <em>H</em> be a closed subgroup of a compact metrizable group <em>G</em>. Then <span><math><mi>G</mi><mo>/</mo><mi>H</mi></math></span> is <em>G</em>-movable if and only if <em>H</em> is a large subgroup of <em>G</em>. It provides a new characterization of large subgroups and generalizes a result of Gevorgyan <span><span>[12]</span></span> about compact Lie groups.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"380 ","pages":"Article 109731"},"PeriodicalIF":0.5,"publicationDate":"2026-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146038150","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 : 2026-01-13DOI: 10.1016/j.topol.2026.109729
Chandrika Sadanand, Emily Shinkle
A subcomplex of a cell complex is called rigid with respect to another cell complex if every injective simplicial map has a unique extension to an injective simplicial map . We say that a cell complex exhibits finite rigidity if it contains a finite, rigid subcomplex. Given a surface with marked points, its flip graph and arc complex are simplicial complexes indexing the triangulations and the arcs between marked points, respectively. In this paper, we leverage the fact that the flip graph can be embedded in the arc complex as its dual to show that finite rigidity of the flip graph implies finite rigidity of the arc complex. Thus, a recent result of the second author on the finite rigidity of the flip graph implies finite rigidity of the arc complex for a broad class of surfaces. Notably, this includes surfaces with boundary – a setting where finite rigidity of the arc complex was previously unknown. We further show that these arc complexes admit exhaustions by finite rigid sets, which was shown to be an important component in the proof of many interesting model-theoretic properties of simplicial complexes associated to surfaces in a recent work of de la Nuez Gonzalez-Disarlo-Koberda.
{"title":"Flip graph and arc complex finite rigidity","authors":"Chandrika Sadanand, Emily Shinkle","doi":"10.1016/j.topol.2026.109729","DOIUrl":"10.1016/j.topol.2026.109729","url":null,"abstract":"<div><div>A subcomplex <span><math><mi>X</mi></math></span> of a cell complex <span><math><mi>C</mi></math></span> is called <em>rigid</em> with respect to another cell complex <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> if every injective simplicial map <span><math><mi>λ</mi><mo>:</mo><mi>X</mi><mo>→</mo><msup><mrow><mi>C</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> has a unique extension to an injective simplicial map <span><math><mi>ϕ</mi><mo>:</mo><mi>C</mi><mo>→</mo><msup><mrow><mi>C</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span>. We say that a cell complex exhibits <em>finite rigidity</em> if it contains a finite, rigid subcomplex. Given a surface with marked points, its <em>flip graph</em> and <em>arc complex</em> are simplicial complexes indexing the triangulations and the arcs between marked points, respectively. In this paper, we leverage the fact that the flip graph can be embedded in the arc complex as its dual to show that finite rigidity of the flip graph implies finite rigidity of the arc complex. Thus, a recent result of the second author on the finite rigidity of the flip graph implies finite rigidity of the arc complex for a broad class of surfaces. Notably, this includes surfaces with boundary – a setting where finite rigidity of the arc complex was previously unknown. We further show that these arc complexes admit exhaustions by finite rigid sets, which was shown to be an important component in the proof of many interesting model-theoretic properties of simplicial complexes associated to surfaces in a recent work of de la Nuez Gonzalez-Disarlo-Koberda.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"381 ","pages":"Article 109729"},"PeriodicalIF":0.5,"publicationDate":"2026-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146080771","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 : 2026-01-13DOI: 10.1016/j.topol.2026.109727
Jacob Mostovoy
We study the space of all configurations of n ordered points on the circle such that no three points coincide, and in which one of the points (say, the last one) is fixed. We compute its fundamental group for and describe its homology for . For arbitrary n, we compute its first homology and its Euler characteristic.
We use three geometric approaches. On one hand, is naturally defined as the complement of an arrangement of codimension-2 subtori in a real torus. On the other hand, is homotopy equivalent to an explicit nonpositively curved cubical complex. Finally, can also be assembled from no-3-equal manifolds of the real line.
We also observe that, up to homotopy, may be identified with a subspace of the oriented double cover of the moduli space of stable real rational curves with n marked points. This gives an embedding of into the pure cactus group. As a corollary, we see that is residually nilpotent.
{"title":"Round twin groups on few strands","authors":"Jacob Mostovoy","doi":"10.1016/j.topol.2026.109727","DOIUrl":"10.1016/j.topol.2026.109727","url":null,"abstract":"<div><div>We study the space <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> of all configurations of <em>n</em> ordered points on the circle such that no three points coincide, and in which one of the points (say, the last one) is fixed. We compute its fundamental group for <span><math><mi>n</mi><mo><</mo><mn>6</mn></math></span> and describe its homology for <span><math><mi>n</mi><mo>=</mo><mn>6</mn><mo>,</mo><mn>7</mn></math></span>. For arbitrary <em>n</em>, we compute its first homology and its Euler characteristic.</div><div>We use three geometric approaches. On one hand, <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is naturally defined as the complement of an arrangement of codimension-2 subtori in a real torus. On the other hand, <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is homotopy equivalent to an explicit nonpositively curved cubical complex. Finally, <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> can also be assembled from no-3-equal manifolds of the real line.</div><div>We also observe that, up to homotopy, <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> may be identified with a subspace of the oriented double cover of the moduli space <span><math><msub><mrow><mover><mrow><mi>M</mi></mrow><mo>‾</mo></mover></mrow><mrow><mn>0</mn><mo>,</mo><mi>n</mi></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math></span> of stable real rational curves with <em>n</em> marked points. This gives an embedding of <span><math><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> into the pure cactus group. As a corollary, we see that <span><math><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is residually nilpotent.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"380 ","pages":"Article 109727"},"PeriodicalIF":0.5,"publicationDate":"2026-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146038152","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 : 2026-01-12DOI: 10.1016/j.topol.2026.109728
Yu-Ming Deng, Liang-Xue Peng
In this note, we prove that every GO-space is s-, which gives an affirmative answer to Lin's question [5, Question 3.1] and Peng's question [6, Question 3.4]. In the last part of this note, we point out that there is a gap in Theorem 3.3 in [6] but the statement of the theorem is correct because in this paper we have established a stronger fact in Theorem 3.4.
{"title":"A note on the local base at closed subsets in GO-spaces","authors":"Yu-Ming Deng, Liang-Xue Peng","doi":"10.1016/j.topol.2026.109728","DOIUrl":"10.1016/j.topol.2026.109728","url":null,"abstract":"<div><div>In this note, we prove that every GO-space is <em>s</em>-<span><math><msub><mrow><mi>m</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>, which gives an affirmative answer to Lin's question <span><span>[5, Question 3.1]</span></span> and Peng's question <span><span>[6, Question 3.4]</span></span>. In the last part of this note, we point out that there is a gap in Theorem 3.3 in <span><span>[6]</span></span> but the statement of the theorem is correct because in this paper we have established a stronger fact in <span><span>Theorem 3.4</span></span>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"380 ","pages":"Article 109728"},"PeriodicalIF":0.5,"publicationDate":"2026-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145978200","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 : 2026-01-12DOI: 10.1016/j.topol.2026.109726
Olena Karlova , Volodymyr Mykhaylyuk
We study the diagonals of (strongly) separately Lipschitz mappings . It is shown that for any metric space X and any normed space Y the diagonals of strongly separately pointwise Lipschitz mappings are exactly stable limits of sequences of pointwise Lipschitz mappings (a mapping on the product of n metric spaces is strongly separately pointwise Lipschitz if it is jointly pointwise Lipschitz mapping with respect to any variables). We introduce classes of mappings between metric spaces X and Y which are recursively defined from pointwise Lipschitz mappings, analogously as mappings of stable Baire classes are recursively defined from continuous mappings. It was shown that for a metric space X and a Banach space Y if and only if there exists a sequence of ambiguous sets of the class n such that every restriction is Lipschitz. Moreover, for any metric space X, any normed space Y and every we construct a separately pointwise Lipschitz mapping with given diagonal .
{"title":"Diagonals of separately pointwise Lipschitz functions of n variables","authors":"Olena Karlova , Volodymyr Mykhaylyuk","doi":"10.1016/j.topol.2026.109726","DOIUrl":"10.1016/j.topol.2026.109726","url":null,"abstract":"<div><div>We study the diagonals <span><math><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>,</mo><mo>…</mo><mo>,</mo><mi>x</mi><mo>)</mo></math></span> of (strongly) separately Lipschitz mappings <span><math><mi>f</mi><mo>:</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>→</mo><mi>Y</mi></math></span>. It is shown that for any metric space <em>X</em> and any normed space <em>Y</em> the diagonals of strongly separately pointwise Lipschitz mappings <span><math><mi>f</mi><mo>:</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>→</mo><mi>Y</mi></math></span> are exactly stable limits of sequences of pointwise Lipschitz mappings (a mapping on the product of <em>n</em> metric spaces is <em>strongly separately pointwise Lipschitz</em> if it is jointly pointwise Lipschitz mapping with respect to any <span><math><mi>n</mi><mo>−</mo><mn>1</mn></math></span> variables). We introduce classes <span><math><msub><mrow><mi>PL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>)</mo></math></span> of mappings between metric spaces <em>X</em> and <em>Y</em> which are recursively defined from pointwise Lipschitz mappings, analogously as mappings of stable Baire classes are recursively defined from continuous mappings. It was shown that <span><math><mi>f</mi><mo>∈</mo><msub><mrow><mi>PL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>)</mo></math></span> for a metric space <em>X</em> and a Banach space <em>Y</em> if and only if there exists a sequence <span><math><msubsup><mrow><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mo>∞</mo></mrow></msubsup></math></span> of ambiguous sets <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>⊆</mo><mi>X</mi></math></span> of the class <em>n</em> such that every restriction <span><math><mi>f</mi><msub><mrow><mo>|</mo></mrow><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msub></math></span> is Lipschitz. Moreover, for any metric space <em>X</em>, any normed space <em>Y</em> and every <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span> we construct a separately pointwise Lipschitz mapping <span><math><mi>f</mi><mo>:</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>→</mo><mi>Y</mi></math></span> with given diagonal <span><math><mi>g</mi><mo>∈</mo><msub><mrow><mi>PL</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"380 ","pages":"Article 109726"},"PeriodicalIF":0.5,"publicationDate":"2026-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145978201","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 : 2026-01-09DOI: 10.1016/j.topol.2026.109725
Debraj Chandra , Nur Alam , Dipika Roy
A space X is H-separable (Bella et al. (2009) [6]) if for every sequence of dense subspaces of X there exists a sequence such that for each n is a finite subset of and every nonempty open set of X intersects for all but finitely many n. In this paper, we introduce and study an ideal variant of H-separability, called -H-separability.
如果对于X的密集子空间的每一个序列（Yn:n∈n）存在一个序列（Fn:n∈n），使得对于每一个n Fn是Yn的有限子集，并且X的每一个非空开集除有限多个n外都与Fn相交，则空间X是h可分的（Bella et al.(2009)[6]）。本文引入并研究了h可分性的一个理想变体，称为i - h可分性。
{"title":"On a variation of selective separability using ideals","authors":"Debraj Chandra , Nur Alam , Dipika Roy","doi":"10.1016/j.topol.2026.109725","DOIUrl":"10.1016/j.topol.2026.109725","url":null,"abstract":"<div><div>A space <em>X</em> is H-separable (Bella et al. (2009) <span><span>[6]</span></span>) if for every sequence <span><math><mo>(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>:</mo><mi>n</mi><mo>∈</mo><mi>N</mi><mo>)</mo></math></span> of dense subspaces of <em>X</em> there exists a sequence <span><math><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>:</mo><mi>n</mi><mo>∈</mo><mi>N</mi><mo>)</mo></math></span> such that for each <em>n</em> <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is a finite subset of <span><math><msub><mrow><mi>Y</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and every nonempty open set of <em>X</em> intersects <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> for all but finitely many <em>n</em>. In this paper, we introduce and study an ideal variant of H-separability, called <span><math><mi>I</mi></math></span>-H-separability.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"380 ","pages":"Article 109725"},"PeriodicalIF":0.5,"publicationDate":"2026-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145978202","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 : 2026-01-08DOI: 10.1016/j.topol.2026.109724
Xuewei Ling
In this paper, we investigate submetrizability in quotient spaces of semitopological groups. The following results are obtained: (1) If H is a closed neutral subgroup of a semitopological group G such that is a Hausdorff paracompact space with , then is submetrizable; (2) If H is a closed neutral subgroup of a semitopological group G such that is Hausdorff (resp., Tychonoff) and , then admits a continuous bijection onto a Hausdorff space with a countable base (resp., admits a weaker separable metrizable topology).
{"title":"Submetrizability in quotient spaces of semitopological groups","authors":"Xuewei Ling","doi":"10.1016/j.topol.2026.109724","DOIUrl":"10.1016/j.topol.2026.109724","url":null,"abstract":"<div><div>In this paper, we investigate submetrizability in quotient spaces of semitopological groups. The following results are obtained: (1) If <em>H</em> is a closed neutral subgroup of a semitopological group <em>G</em> such that <span><math><mi>G</mi><mo>/</mo><mi>H</mi></math></span> is a Hausdorff paracompact space with <span><math><mi>H</mi><msub><mrow><mi>s</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>/</mo><mi>H</mi><mo>)</mo><mo>⋅</mo><mi>ψ</mi><mo>(</mo><mi>G</mi><mo>/</mo><mi>H</mi><mo>)</mo><mo>≤</mo><mi>ω</mi></math></span>, then <span><math><mi>G</mi><mo>/</mo><mi>H</mi></math></span> is submetrizable; (2) If <em>H</em> is a closed neutral subgroup of a semitopological group <em>G</em> such that <span><math><mi>G</mi><mo>/</mo><mi>H</mi></math></span> is Hausdorff (resp., Tychonoff) and <span><math><mi>H</mi><msub><mrow><mi>s</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>/</mo><mi>H</mi><mo>)</mo><mo>⋅</mo><mi>I</mi><msub><mrow><mi>n</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>/</mo><mi>H</mi><mo>)</mo><mo>⋅</mo><mi>ψ</mi><mo>(</mo><mi>G</mi><mo>/</mo><mi>H</mi><mo>)</mo><mo>≤</mo><mi>ω</mi></math></span>, then <span><math><mi>G</mi><mo>/</mo><mi>H</mi></math></span> admits a continuous bijection onto a Hausdorff space with a countable base (resp., admits a weaker separable metrizable topology).</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"380 ","pages":"Article 109724"},"PeriodicalIF":0.5,"publicationDate":"2026-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145978199","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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