Pub Date : 2026-01-01Epub Date: 2025-10-10DOI: 10.1016/j.topol.2025.109635
Ajit Kumar Gupta , Saikat Mukherjee
We define two properties for subsets of a metric space. One of them is a generalization of chainability, finite chainability, and Menger convexity for metric spaces; and the other extends the notion of compactness for subsets of a metric space. We establish several fundamental results concerning these two properties. Further, in the context of these properties, we study the Hausdorff metric and derive the relations among Hausdorff, Vietoris, and locally finite hypertopologies on the collection of nonempty closed subsets of a metric space.
{"title":"Generalizations of chainability and compactness, and the hypertopologies","authors":"Ajit Kumar Gupta , Saikat Mukherjee","doi":"10.1016/j.topol.2025.109635","DOIUrl":"10.1016/j.topol.2025.109635","url":null,"abstract":"<div><div>We define two properties for subsets of a metric space. One of them is a generalization of chainability, finite chainability, and Menger convexity for metric spaces; and the other extends the notion of compactness for subsets of a metric space. We establish several fundamental results concerning these two properties. Further, in the context of these properties, we study the Hausdorff metric and derive the relations among Hausdorff, Vietoris, and locally finite hypertopologies on the collection of nonempty closed subsets of a metric space.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109635"},"PeriodicalIF":0.5,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145321824","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: 2025-10-17DOI: 10.1016/j.topol.2025.109639
Jing Song , Meng Bao , Xiaolan Liu , Xuewei Ling
A topological gyrogroup is a gyrogroup endowed with a topology such that the binary operation is jointly continuous and the inverse mapping is also continuous. In this paper, it is shown that a strongly topological gyrogroup G is strongly countably complete if and only if G contains a closed countably compact strong subgyrogroup H such that the quotient space is completely metrizable and the canonical quotient mapping is closed, which gives an affirmative answer to [40, Question 4.11] and it deduces that a strongly topological gyrogroup G is strongly countably complete if and only if it is countably sieve-complete. Then it is claimed that every symmetrizable Hausdorff strongly paratopological gyrogroup with the Baire property is a metrizable strongly topological gyrogroup.
{"title":"Some characterizations on strongly topological gyrogroups","authors":"Jing Song , Meng Bao , Xiaolan Liu , Xuewei Ling","doi":"10.1016/j.topol.2025.109639","DOIUrl":"10.1016/j.topol.2025.109639","url":null,"abstract":"<div><div>A topological gyrogroup is a gyrogroup endowed with a topology such that the binary operation is jointly continuous and the inverse mapping is also continuous. In this paper, it is shown that a strongly topological gyrogroup <em>G</em> is strongly countably complete if and only if <em>G</em> contains a closed countably compact strong subgyrogroup <em>H</em> such that the quotient space <span><math><mi>G</mi><mo>/</mo><mi>H</mi></math></span> is completely metrizable and the canonical quotient mapping <span><math><mi>π</mi><mo>:</mo><mi>G</mi><mo>→</mo><mi>G</mi><mo>/</mo><mi>H</mi></math></span> is closed, which gives an affirmative answer to <span><span>[40, Question 4.11]</span></span> and it deduces that a strongly topological gyrogroup <em>G</em> is strongly countably complete if and only if it is countably sieve-complete. Then it is claimed that every symmetrizable Hausdorff strongly paratopological gyrogroup with the Baire property is a metrizable strongly topological gyrogroup.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109639"},"PeriodicalIF":0.5,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145418098","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: 2025-10-13DOI: 10.1016/j.topol.2025.109636
Khulod Almontashery , Paul J. Szeptycki
We introduce a strengthening of the class of the proximal and semi-proximal spaces by restricting the proximal game to totally bounded uniformities. In addition, we examine the connections between the proximal game and two well-known games, one set-theoretic the other topological: the Galvin game and the Gruenhage game.
{"title":"The proximal game and its connections to other games","authors":"Khulod Almontashery , Paul J. Szeptycki","doi":"10.1016/j.topol.2025.109636","DOIUrl":"10.1016/j.topol.2025.109636","url":null,"abstract":"<div><div>We introduce a strengthening of the class of the proximal and semi-proximal spaces by restricting the proximal game to totally bounded uniformities. In addition, we examine the connections between the proximal game and two well-known games, one set-theoretic the other topological: the Galvin game and the Gruenhage game.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109636"},"PeriodicalIF":0.5,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145321827","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: 2025-10-27DOI: 10.1016/j.topol.2025.109648
Yanhui Huang
In this paper, we discuss the relationships among -compactness, star countability, star Lindelöfness, star almost Lindelöfness and star weakly Lindelöfness in different spaces. We mainly give the following:
(1)
For a subspace X of an ordinal, X is star weakly Lindelöf if and only if it is -compact.
(2)
For subspaces A and B of an ordinal, is star weakly Lindelöf if and only if it is -compact.
(3)
For a subspace X of , X is star weakly Lindelöf if and only if it is -compact.
{"title":"Star covering properties of products of subspaces of ordinals","authors":"Yanhui Huang","doi":"10.1016/j.topol.2025.109648","DOIUrl":"10.1016/j.topol.2025.109648","url":null,"abstract":"<div><div>In this paper, we discuss the relationships among <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-compactness, star countability, star Lindelöfness, star almost Lindelöfness and star weakly Lindelöfness in different spaces. We mainly give the following:<ul><li><span>(1)</span><span><div>For a subspace <em>X</em> of an ordinal, <em>X</em> is star weakly Lindelöf if and only if it is <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-compact.</div></span></li><li><span>(2)</span><span><div>For subspaces <em>A</em> and <em>B</em> of an ordinal, <span><math><mi>A</mi><mo>×</mo><mi>B</mi></math></span> is star weakly Lindelöf if and only if it is <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-compact.</div></span></li><li><span>(3)</span><span><div>For a subspace <em>X</em> of <span><math><msubsup><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup></math></span>, <em>X</em> is star weakly Lindelöf if and only if it is <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-compact.</div></span></li></ul></div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109648"},"PeriodicalIF":0.5,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145418008","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: 2025-10-20DOI: 10.1016/j.topol.2025.109640
Evgenii Reznichenko, Ol'ga Sipacheva
The problem of the existence of non-pseudo--compact -factorizable groups is studied. It is proved that any such group is submetrizable and has weight larger than . Closely related results concerning the -factorizability of products of topological groups and spaces are also obtained (a product of topological spaces is said to be -factorizable if any continuous function factors through a product of maps from X and Y to second-countable spaces). In particular, it is proved that the square of a topological group G is -factorizable as a group if and only if it is -factorizable as a product of spaces, in which case G is pseudo--compact. It is also proved that if the product of a space X and an uncountable discrete space is -factorizable, then is hereditarily separable and hereditarily Lindelöf.
{"title":"Weird R-factorizable groups","authors":"Evgenii Reznichenko, Ol'ga Sipacheva","doi":"10.1016/j.topol.2025.109640","DOIUrl":"10.1016/j.topol.2025.109640","url":null,"abstract":"<div><div>The problem of the existence of non-pseudo-<span><math><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-compact <span><math><mi>R</mi></math></span>-factorizable groups is studied. It is proved that any such group is submetrizable and has weight larger than <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>. Closely related results concerning the <span><math><mi>R</mi></math></span>-factorizability of products of topological groups and spaces are also obtained (a product <span><math><mi>X</mi><mo>×</mo><mi>Y</mi></math></span> of topological spaces is said to be <span><math><mi>R</mi></math></span>-factorizable if any continuous function <span><math><mi>X</mi><mo>×</mo><mi>Y</mi><mo>→</mo><mi>R</mi></math></span> factors through a product of maps from <em>X</em> and <em>Y</em> to second-countable spaces). In particular, it is proved that the square <span><math><mi>G</mi><mo>×</mo><mi>G</mi></math></span> of a topological group <em>G</em> is <span><math><mi>R</mi></math></span>-factorizable as a group if and only if it is <span><math><mi>R</mi></math></span>-factorizable as a product of spaces, in which case <em>G</em> is pseudo-<span><math><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-compact. It is also proved that if the product of a space <em>X</em> and an uncountable discrete space is <span><math><mi>R</mi></math></span>-factorizable, then <span><math><msup><mrow><mi>X</mi></mrow><mrow><mi>ω</mi></mrow></msup></math></span> is hereditarily separable and hereditarily Lindelöf.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109640"},"PeriodicalIF":0.5,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145364654","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: 2025-09-12DOI: 10.1016/j.topol.2025.109597
Silvère Gangloff , Pierre Guillon , Piotr Oprocha
It is well-known that when a positively expansive dynamical system is invertible, its underlying space is finite. C. Morales has introduced a decade ago a natural way to generalize positive expansiveness, by introducing other properties that he called positive n-expansiveness, for all , positive 1-expansiveness being identical to positive expansiveness. Contrary to positive expansiveness, positive n-expansiveness for does not enforce that the space is finite when the system is invertible. In the present paper we call finitely positively expansive dynamical systems as the ones which are positively n-expansive for some integer n, and prove several results on this class of systems. In particular, the well-known result quoted above is true if we add the constraint of shadowing property, while it is not if this property is replaced with minimality. Furthermore, finitely positively expansive systems cannot occur on certain topological spaces such as the interval, when the system is assumed to be invertible finite positive expansiveness implies zero topological entropy. Overall we show that the class of finitely positively expansive dynamical systems is quite rich and leave several questions open for further research.
{"title":"Various questions around finitely positively expansive dynamical systems","authors":"Silvère Gangloff , Pierre Guillon , Piotr Oprocha","doi":"10.1016/j.topol.2025.109597","DOIUrl":"10.1016/j.topol.2025.109597","url":null,"abstract":"<div><div>It is well-known that when a positively expansive dynamical system is invertible, its underlying space is finite. C. Morales has introduced a decade ago a natural way to generalize positive expansiveness, by introducing other properties that he called positive <em>n</em>-expansiveness, for all <span><math><mi>n</mi><mo>≥</mo><mn>1</mn></math></span>, positive 1-expansiveness being identical to positive expansiveness. Contrary to positive expansiveness, positive <em>n</em>-expansiveness for <span><math><mi>n</mi><mo>></mo><mn>1</mn></math></span> does not enforce that the space is finite when the system is invertible. In the present paper we call finitely positively expansive dynamical systems as the ones which are positively <em>n</em>-expansive for some integer <em>n</em>, and prove several results on this class of systems. In particular, the well-known result quoted above is true if we add the constraint of shadowing property, while it is not if this property is replaced with minimality. Furthermore, finitely positively expansive systems cannot occur on certain topological spaces such as the interval, when the system is assumed to be invertible finite positive expansiveness implies zero topological entropy. Overall we show that the class of finitely positively expansive dynamical systems is quite rich and leave several questions open for further research.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109597"},"PeriodicalIF":0.5,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145322261","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.2025.109633
Yongqiang Liu , Wentao Xie
In this paper, we study the first homology group of finite cyclic covering of complex line arrangement complement. We show that this first integral homology group is torsion-free under certain condition similar to the one used by Cohen-Dimca-Orlik [3, Theorem 1]. In particular, this includes the case of the Milnor fiber, which generalizes the previous results obtained by Williams [36, Main Theorem 1] for complexified line arrangement to any complex line arrangement.
本文研究了复线排列补的有限循环覆盖的第一同调群。在与Cohen-Dimca-Orlik[3，定理1]相似的条件下，证明了第一个积分同调群是无扭转的。特别地，这包括Milnor纤维的情况，它将Williams [36， Main Theorem 1]先前得到的关于复线排列的结果推广到任何复线排列。
{"title":"The homology groups of finite cyclic coverings of line arrangement complements","authors":"Yongqiang Liu , Wentao Xie","doi":"10.1016/j.topol.2025.109633","DOIUrl":"10.1016/j.topol.2025.109633","url":null,"abstract":"<div><div>In this paper, we study the first homology group of finite cyclic covering of complex line arrangement complement. We show that this first integral homology group is torsion-free under certain condition similar to the one used by Cohen-Dimca-Orlik <span><span>[3, Theorem 1]</span></span>. In particular, this includes the case of the Milnor fiber, which generalizes the previous results obtained by Williams <span><span>[36, Main Theorem 1]</span></span> for complexified line arrangement to any complex line arrangement.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109633"},"PeriodicalIF":0.5,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145321825","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: 2025-10-27DOI: 10.1016/j.topol.2025.109647
J.C. Ferrando , J. Ka̧kol
Let be the linear space of real-valued continuous functions with the pointwise topology. It is known that a Tychonoff space X is a Δ-space if and only if the locally convex space is distinguished. It has been recently shown that if there is a continuous linear surjection from onto and X is a Δ-space, Y is also a Δ-space. Here we investigate under what conditions the presence of a dense distinguished subspace E in leads X to be a Δ-space. We also produce a class of spaces for which contains a distinguished dense subspace.
{"title":"Distinguished dense Cp-subspaces","authors":"J.C. Ferrando , J. Ka̧kol","doi":"10.1016/j.topol.2025.109647","DOIUrl":"10.1016/j.topol.2025.109647","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>p</mi></mrow></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></math></span> be the linear space of real-valued continuous functions with the pointwise topology. It is known that a Tychonoff space <em>X</em> is a Δ-space if and only if the locally convex space <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is distinguished. It has been recently shown that if there is a continuous linear surjection from <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> onto <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>Y</mi><mo>)</mo></math></span> and <em>X</em> is a Δ-space, <em>Y</em> is also a Δ-space. Here we investigate under what conditions the presence of a dense distinguished subspace <em>E</em> in <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>p</mi></mrow></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></math></span> leads <em>X</em> to be a Δ-space. We also produce a class of spaces <span><math><mi>X</mi><mo>∉</mo><mi>Δ</mi></math></span> for which <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>p</mi></mrow></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></math></span> contains a distinguished dense subspace.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109647"},"PeriodicalIF":0.5,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145418009","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: 2025-09-24DOI: 10.1016/j.topol.2025.109599
Kaori Yamazaki
In this paper, we show that, for an increasing bounded uniformly continuous function f on a subspace A of a uniform space X equipped with a preorder, f can be extended to an increasing uniformly continuous function on X if and only if f is uniformly completely order separated in X. This extends McShane's Extension Theorem for metric spaces and Katětov's Theorem for uniform spaces. Moreover, we establish a characterization of a uniform/metric space X equipped with a preorder possessing the monotone uniform extension property, which answers a question asked by E.A.Ok.
{"title":"Extensions of increasing bounded uniformly continuous functions","authors":"Kaori Yamazaki","doi":"10.1016/j.topol.2025.109599","DOIUrl":"10.1016/j.topol.2025.109599","url":null,"abstract":"<div><div>In this paper, we show that, for an increasing bounded uniformly continuous function <em>f</em> on a subspace <em>A</em> of a uniform space <em>X</em> equipped with a preorder, <em>f</em> can be extended to an increasing uniformly continuous function on <em>X</em> if and only if <em>f</em> is uniformly completely order separated in <em>X</em>. This extends McShane's Extension Theorem for metric spaces and Katětov's Theorem for uniform spaces. Moreover, we establish a characterization of a uniform/metric space <em>X</em> equipped with a preorder possessing the monotone uniform extension property, which answers a question asked by E.A.Ok.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109599"},"PeriodicalIF":0.5,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145222985","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.2025.109622
Ryuji Higa
We consider the problem of determining whether two given virtual knots can be converted into each other by a sequence of crossing changes or a sequence of Δ-moves. We provide a simple method derived from the r-covering of a virtual knot for approaching this problem. We also give the lower bounds of the Gordian distance for crossing changes and Δ-moves.
{"title":"A note on crossing changes and delta-moves for virtual knots","authors":"Ryuji Higa","doi":"10.1016/j.topol.2025.109622","DOIUrl":"10.1016/j.topol.2025.109622","url":null,"abstract":"<div><div>We consider the problem of determining whether two given virtual knots can be converted into each other by a sequence of crossing changes or a sequence of Δ-moves. We provide a simple method derived from the <em>r</em>-covering of a virtual knot for approaching this problem. We also give the lower bounds of the Gordian distance for crossing changes and Δ-moves.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"377 ","pages":"Article 109622"},"PeriodicalIF":0.5,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145223046","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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