Pub Date : 2026-01-01Epub Date: 2025-10-28DOI: 10.1016/j.topol.2025.109653
Nathan Carlson
We give a new bound for the cardinality of a Tychonoff homogeneous space using cozero sets. This leads to improved cardinal inequalities for compact homogeneous spaces that generalize to the locally compact setting. In this connection it is also shown that for any Hausdorff space X, where is the point-wise compactness type of X. This extends Arhangel′skiĭ's result that when X is compact Hausdorff. In addition pseudocompactness is investigated in connection with homogeneity. Among other results, we show that if X is a ccc locally compact noncompact space such that the one-point compactification of X is homogeneous and has character , then X is pseudocompact. It follows that if X is either or and then is pseudocompact.
{"title":"On local compactness, pseudocompactness, and homogeneity","authors":"Nathan Carlson","doi":"10.1016/j.topol.2025.109653","DOIUrl":"10.1016/j.topol.2025.109653","url":null,"abstract":"<div><div>We give a new bound for the cardinality of a Tychonoff homogeneous space using cozero sets. This leads to improved cardinal inequalities for compact homogeneous spaces that generalize to the locally compact setting. In this connection it is also shown that <span><math><mi>w</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>≤</mo><mi>n</mi><mi>w</mi><msup><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mrow><mi>p</mi><mi>c</mi><mi>t</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msup></math></span> for any Hausdorff space <em>X</em>, where <span><math><mi>p</mi><mi>c</mi><mi>t</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is the point-wise compactness type of <em>X</em>. This extends Arhangel′skiĭ's result that <span><math><mi>w</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><mi>n</mi><mi>w</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> when <em>X</em> is compact Hausdorff. In addition pseudocompactness is investigated in connection with homogeneity. Among other results, we show that if <em>X</em> is a ccc locally compact noncompact space such that the one-point compactification of <em>X</em> is homogeneous and has character <span><math><mi>c</mi></math></span>, then <em>X</em> is pseudocompact. It follows that if <em>X</em> is either <span><math><msup><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mrow><mi>c</mi></mrow></msup></math></span> or <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>c</mi></mrow></msup></math></span> and <span><math><mi>p</mi><mo>∈</mo><mi>X</mi></math></span> then <span><math><mi>X</mi><mo>﹨</mo><mo>{</mo><mi>p</mi><mo>}</mo></math></span> is pseudocompact.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109653"},"PeriodicalIF":0.5,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145418097","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 : 2026-01-01Epub Date: 2025-10-30DOI: 10.1016/j.topol.2025.109652
Ashwani K B, Ali Akbar K
A chaotic group action is a nonminimal, topologically transitive continuous group action with dense periodic points. In this paper, we discuss indecomposability for a continuous group action and prove that indecomposability is an equivalent definition of topological transitivity. Moreover, we prove that any infinite compact subset of the real line having a chaotic group action is homeomorphic to the middle third Cantor set.
{"title":"Indecomposability of group actions","authors":"Ashwani K B, Ali Akbar K","doi":"10.1016/j.topol.2025.109652","DOIUrl":"10.1016/j.topol.2025.109652","url":null,"abstract":"<div><div>A chaotic group action is a nonminimal, topologically transitive continuous group action with dense periodic points. In this paper, we discuss indecomposability for a continuous group action and prove that indecomposability is an equivalent definition of topological transitivity. Moreover, we prove that any infinite compact subset of the real line having a chaotic group action is homeomorphic to the middle third Cantor set.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109652"},"PeriodicalIF":0.5,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145466892","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 : 2026-01-01Epub Date: 2025-10-31DOI: 10.1016/j.topol.2025.109657
Kendall Heiney , Margaret Kipe , Samantha Pezzimenti , Kaelyn Pontes , Lực Ta
Knot mosaics were introduced by Kauffman and Lomonaco in the context of quantum knots, but have since been studied for their own right. A classical knot mosaic is formed on a square grid. In this work, we identify opposite edges of the square to form mosaics on the surface of a torus. We provide two algorithms for efficiently constructing toric mosaics of torus knots, providing upper bounds for the toric mosaic number. Using these results and a computer search, we provide a census of known toric mosaic numbers.
{"title":"Constructions of and bounds on the toric mosaic number","authors":"Kendall Heiney , Margaret Kipe , Samantha Pezzimenti , Kaelyn Pontes , Lực Ta","doi":"10.1016/j.topol.2025.109657","DOIUrl":"10.1016/j.topol.2025.109657","url":null,"abstract":"<div><div>Knot mosaics were introduced by Kauffman and Lomonaco in the context of quantum knots, but have since been studied for their own right. A classical knot mosaic is formed on a square grid. In this work, we identify opposite edges of the square to form mosaics on the surface of a torus. We provide two algorithms for efficiently constructing toric mosaics of torus knots, providing upper bounds for the toric mosaic number. Using these results and a computer search, we provide a census of known toric mosaic numbers.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109657"},"PeriodicalIF":0.5,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145466891","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 : 2026-01-01Epub Date: 2025-11-11DOI: 10.1016/j.topol.2025.109658
Danica Kosanović
Motivated by recent results on diffeomorphisms of 4-manifolds, this paper investigates fundamental groups of spaces of embeddings of in 4-manifolds. The majority of work goes into the case of framed immersed circles.
{"title":"On fundamental groups of spaces of framed embeddings of a circle in a 4-manifold","authors":"Danica Kosanović","doi":"10.1016/j.topol.2025.109658","DOIUrl":"10.1016/j.topol.2025.109658","url":null,"abstract":"<div><div>Motivated by recent results on diffeomorphisms of 4-manifolds, this paper investigates fundamental groups of spaces of embeddings of <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>×</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> in 4-manifolds. The majority of work goes into the case of framed immersed circles.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109658"},"PeriodicalIF":0.5,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145520225","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 : 2026-01-01Epub Date: 2025-10-02DOI: 10.1016/j.topol.2025.109601
Mengqiao Huang , Xiaodong Jia , Qingguo Li
For a weak partial metric space , there is a canonical metric on X, defined as for all . We prove that the partial metric topology and the Scott topology on coincide if and only if the metric topology on and the Lawson topology on agree, provided that the weak partial metric space is a domain in its specialization order and its associated metric space is compact. We also discussed fixpoints of self maps defined on weak partial metric spaces.
{"title":"Topologies and fixpoints on weak partial metric spaces","authors":"Mengqiao Huang , Xiaodong Jia , Qingguo Li","doi":"10.1016/j.topol.2025.109601","DOIUrl":"10.1016/j.topol.2025.109601","url":null,"abstract":"<div><div>For a weak partial metric space <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>p</mi><mo>)</mo></math></span>, there is a canonical metric <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> on <em>X</em>, defined as <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>=</mo><mi>max</mi><mo></mo><mo>{</mo><mi>p</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>−</mo><mi>p</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo>)</mo><mo>,</mo><mi>p</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>−</mo><mi>p</mi><mo>(</mo><mi>y</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>}</mo></math></span> for all <span><math><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>X</mi></math></span>. We prove that the partial metric topology and the Scott topology on <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>p</mi><mo>)</mo></math></span> coincide if and only if the metric topology on <span><math><mo>(</mo><mi>X</mi><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></math></span> and the Lawson topology on <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>p</mi><mo>)</mo></math></span> agree, provided that the weak partial metric space <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>p</mi><mo>)</mo></math></span> is a domain in its specialization order and its associated metric space <span><math><mo>(</mo><mi>X</mi><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></math></span> is compact. We also discussed fixpoints of self maps defined on weak partial metric spaces.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109601"},"PeriodicalIF":0.5,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145269836","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 : 2026-01-01Epub Date: 2025-09-30DOI: 10.1016/j.topol.2025.109603
Chi-Heng Zhang , Nan Gao , Zi-Cheng Cheng
Gabrel-Krause dimension of the rational cohomology is described for the m-torus . Inspired by the diagonalizability of admissible map between , the relationship of minimal realization among symmetrizable generalised Cartan matrices is shown.
{"title":"Rational cohomology and Cartan matrix","authors":"Chi-Heng Zhang , Nan Gao , Zi-Cheng Cheng","doi":"10.1016/j.topol.2025.109603","DOIUrl":"10.1016/j.topol.2025.109603","url":null,"abstract":"<div><div>Gabrel-Krause dimension of the rational cohomology <span><math><msup><mrow><mi>H</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><mi>B</mi><msup><mrow><mi>T</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>;</mo><mi>Q</mi><mo>)</mo></math></span> is described for the <em>m</em>-torus <span><math><msup><mrow><mi>T</mi></mrow><mrow><mi>m</mi></mrow></msup></math></span>. Inspired by the diagonalizability of admissible map between <span><math><msup><mrow><mi>H</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><mi>B</mi><msup><mrow><mi>T</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>,</mo><mi>Q</mi><mo>)</mo></math></span>, the relationship of minimal realization among symmetrizable generalised Cartan matrices is shown.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109603"},"PeriodicalIF":0.5,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145269841","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 : 2026-01-01Epub Date: 2025-09-03DOI: 10.1016/j.topol.2025.109574
Jeffrey T. Denniston , Stephen E. Rodabaugh , Jamal K. Tartir
This paper focuses on the Kolmogorov functor and associated ideas. There are two main objectives: first, catalogue and prove those topological invariants which K both preserves and reflects, called “Hong” invariants; and second, give a step-by-step axiomatic foundation for K to analyze its remarkable success in having so many Hong invariants. Pursuing the second objective leads to “essentially Kolmogorov” (EK) relations, the family of which on a ground set forms a complete lattice ordered by inclusion; the diagonal relation Δ is the universal lower bound and the Kolmogorov relation K is the universal upper bound—typically there are many EK relations strictly between Δ and K. Though EK relations are significant weakenings of K, they enjoy the same success w.r.t. Hong invariants. Counterexamples clarify relationships between similar notions.
{"title":"Preservation and reflection of separation axioms by essentially Kolmogorov and Kolmogorov relations","authors":"Jeffrey T. Denniston , Stephen E. Rodabaugh , Jamal K. Tartir","doi":"10.1016/j.topol.2025.109574","DOIUrl":"10.1016/j.topol.2025.109574","url":null,"abstract":"<div><div>This paper focuses on the Kolmogorov functor <span><math><mi>K</mi><mo>:</mo><mrow><mi>Top</mi></mrow><mo>→</mo><msub><mrow><mi>Top</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> and associated ideas. There are two main objectives: first, catalogue and prove those topological invariants which <em>K</em> both preserves and reflects, called “Hong” invariants; and second, give a step-by-step axiomatic foundation for <em>K</em> to analyze its remarkable success in having so many Hong invariants. Pursuing the second objective leads to “essentially Kolmogorov” (EK) relations, the family of which on a ground set forms a complete lattice ordered by inclusion; the diagonal relation Δ is the universal lower bound and the Kolmogorov relation <em>K</em> is the universal upper bound—typically there are many EK relations strictly between Δ and <em>K</em>. Though EK relations are significant weakenings of <em>K</em>, they enjoy the same success w.r.t. Hong invariants. Counterexamples clarify relationships between similar notions.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109574"},"PeriodicalIF":0.5,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145222986","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 : 2026-01-01Epub Date: 2025-09-25DOI: 10.1016/j.topol.2025.109600
Abolfazl Tarizadeh
If R is a topological ring then , the group of units of R, with the subspace topology is not necessarily a topological group. This leads us to the following natural definition: By absolute topological ring we mean a topological ring such that its group of units with the subspace topology is a topological group. We prove that every commutative ring with the I-adic topology is an absolute topological ring (where I is an ideal of the ring).
Next, we prove that if I is an ideal of a ring R then for the I-adic topology over R we have where is the space of connected components of R and is the space of irreducible closed subsets of R.
We also show with an example that the identity component of a topological group is not necessarily a characteristic subgroup.
Finally, we observed that the main result of Koh [3] as well as its corrected form [5, Chap II, §12, Theorem 12.1] is not true, and then we corrected this result in the right way.
{"title":"Some notes on topological rings and their groups of units","authors":"Abolfazl Tarizadeh","doi":"10.1016/j.topol.2025.109600","DOIUrl":"10.1016/j.topol.2025.109600","url":null,"abstract":"<div><div>If <em>R</em> is a topological ring then <span><math><msup><mrow><mi>R</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>, the group of units of <em>R</em>, with the subspace topology is not necessarily a topological group. This leads us to the following natural definition: By absolute topological ring we mean a topological ring such that its group of units with the subspace topology is a topological group. We prove that every commutative ring with the <em>I</em>-adic topology is an absolute topological ring (where <em>I</em> is an ideal of the ring).</div><div>Next, we prove that if <em>I</em> is an ideal of a ring <em>R</em> then for the <em>I</em>-adic topology over <em>R</em> we have <span><math><msub><mrow><mi>π</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo><mo>=</mo><mi>R</mi><mo>/</mo><mo>(</mo><munder><mo>⋂</mo><mrow><mi>n</mi><mo>⩾</mo><mn>1</mn></mrow></munder><msup><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo><mo>=</mo><mi>t</mi><mo>(</mo><mi>R</mi><mo>)</mo></math></span> where <span><math><msub><mrow><mi>π</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math></span> is the space of connected components of <em>R</em> and <span><math><mi>t</mi><mo>(</mo><mi>R</mi><mo>)</mo></math></span> is the space of irreducible closed subsets of <em>R</em>.</div><div>We also show with an example that the identity component of a topological group is not necessarily a characteristic subgroup.</div><div>Finally, we observed that the main result of Koh <span><span>[3]</span></span> as well as its corrected form <span><span>[5, Chap II, §12, Theorem 12.1]</span></span> is not true, and then we corrected this result in the right way.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109600"},"PeriodicalIF":0.5,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145183877","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 : 2026-01-01Epub Date: 2025-11-26DOI: 10.1016/j.topol.2025.109666
Shou Lin , Ying Ge , Xiangeng Zhou
This note provides a corrigendum to the proof of Theorem 4.5 in Topol. Appl. 271 (2020) 107049.
本注释提供了对Topol中定理4.5的证明的更正。应用程序271(2020)107049。
{"title":"Corrigendum to “Compact-star networks and the images of metric spaces under C-mappings” [Topol. Appl. 271 (2020) 107049]","authors":"Shou Lin , Ying Ge , Xiangeng Zhou","doi":"10.1016/j.topol.2025.109666","DOIUrl":"10.1016/j.topol.2025.109666","url":null,"abstract":"<div><div>This note provides a corrigendum to the proof of Theorem 4.5 in Topol. Appl. 271 (2020) 107049.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109666"},"PeriodicalIF":0.5,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145617797","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 : 2026-01-01Epub Date: 2025-10-10DOI: 10.1016/j.topol.2025.109634
Hadi Hassanzada, Hamid Torabi, Hanieh Mirebrahimi, Ameneh Babaee
In this paper, we present a framework for discrete motion planning tailored for robots that operate in a discrete manner. Furthermore, we extend the concept of r-discrete homotopy as discrete -homotopy. Utilizing this framework, we investigate the notion of discrete topological complexity, which is defined as the least number of motion planning algorithms necessary for discrete movement. We establish several properties related to discrete topological complexity; for example, we demonstrate that discrete motion planning within a metric space X is feasible if and only if X is a discrete contractible space. Additionally, we show that the discrete topological complexity is solely determined by the strictly discrete homotopy type of the spaces involved.
{"title":"A discrete topological complexity of discrete motion planning","authors":"Hadi Hassanzada, Hamid Torabi, Hanieh Mirebrahimi, Ameneh Babaee","doi":"10.1016/j.topol.2025.109634","DOIUrl":"10.1016/j.topol.2025.109634","url":null,"abstract":"<div><div>In this paper, we present a framework for discrete motion planning tailored for robots that operate in a discrete manner. Furthermore, we extend the concept of <em>r</em>-discrete homotopy as discrete <span><math><mo>(</mo><mi>s</mi><mo>,</mo><mi>r</mi><mo>)</mo></math></span>-homotopy. Utilizing this framework, we investigate the notion of discrete topological complexity, which is defined as the least number of motion planning algorithms necessary for discrete movement. We establish several properties related to discrete topological complexity; for example, we demonstrate that discrete motion planning within a metric space <em>X</em> is feasible if and only if <em>X</em> is a discrete contractible space. Additionally, we show that the discrete topological complexity is solely determined by the strictly discrete homotopy type of the spaces involved.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109634"},"PeriodicalIF":0.5,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145269844","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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