Guojun WuSchool of Mathematics and Statistics, Nanjing University of Information Science and TechnologyApplied Mathematics Center of Jiangsu Province, Nanjing University of Information Science and Technology, Wei YaoSchool of Mathematics and Statistics, Nanjing University of Information Science and TechnologyApplied Mathematics Center of Jiangsu Province, Nanjing University of Information Science and Technology, Qingguo LiSchool of Mathematics, Hunan University
With a frame $L$ as the truth value table, we study the topological representations for frame-valued domains. We introduce the notions of locally super-compact $L$-topological space and strong locally super-compact $L$-topological space. Using these concepts, continuous $L$-dcpos and algebraic $L$-dcpos are successfully represented via $L$-sobriety. By means of Scott $L$-topology and specialization $L$-order, we establish a categorical isomorphism between the category of the continuous (resp., algebraic) $L$-dcpos with Scott continuous maps and that of the locally super-compact (resp., strong locally super-compact) $L$-sober spaces with continuous maps. As an application, for a continuous $L$-poset $P$, we obtain a categorical isomorphism between the category of directed completions of $P$ with Scott continuous maps and that of the $L$-sobrifications of $(P, sigma_{L}(P))$ with continuous maps.
{"title":"Topological representations for frame-valued domains via $L$-sobriety","authors":"Guojun WuSchool of Mathematics and Statistics, Nanjing University of Information Science and TechnologyApplied Mathematics Center of Jiangsu Province, Nanjing University of Information Science and Technology, Wei YaoSchool of Mathematics and Statistics, Nanjing University of Information Science and TechnologyApplied Mathematics Center of Jiangsu Province, Nanjing University of Information Science and Technology, Qingguo LiSchool of Mathematics, Hunan University","doi":"arxiv-2406.13595","DOIUrl":"https://doi.org/arxiv-2406.13595","url":null,"abstract":"With a frame $L$ as the truth value table, we study the topological\u0000representations for frame-valued domains. We introduce the notions of locally\u0000super-compact $L$-topological space and strong locally super-compact\u0000$L$-topological space. Using these concepts, continuous $L$-dcpos and algebraic\u0000$L$-dcpos are successfully represented via $L$-sobriety. By means of Scott\u0000$L$-topology and specialization $L$-order, we establish a categorical\u0000isomorphism between the category of the continuous (resp., algebraic) $L$-dcpos\u0000with Scott continuous maps and that of the locally super-compact (resp., strong\u0000locally super-compact) $L$-sober spaces with continuous maps. As an\u0000application, for a continuous $L$-poset $P$, we obtain a categorical\u0000isomorphism between the category of directed completions of $P$ with Scott\u0000continuous maps and that of the $L$-sobrifications of $(P, sigma_{L}(P))$ with\u0000continuous maps.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"208 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141524744","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Alexander R. Klotz, Caleb J. Anderson, Michael S. Dimitriyev
Motivated by the observation of positive Gaussian curvature in kinetoplast DNA networks, we consider the effect of linking chirality in square lattice molecular chainmail networks using Langevin dynamics simulations and constrained gradient optimization. Linking chirality here refers to ordering of over-under versus under-over linkages between a loop and its neighbors. We consider fully alternating linking, maximally non-alternating, and partially non-alternating linking chiralities. We find that in simulations of polymer chainmail networks, the linking chirality dictates the sign of the Gaussian curvature of the final state of the chainmail membranes. Alternating networks have positive Gaussian curvature, similar to what is observed in kinetoplast DNA networks. Maximally non-alternating networks form isotropic membranes with negative Gaussian curvature. Partially non-alternating networks form flat diamond-shaped sheets which undergo a thermal folding transition when sufficiently large, similar to the crumpling transition in tethered membranes. We further investigate this topology-curvature relationship on geometric grounds by considering the tightest possible configurations and the constraints that must be satisfied to achieve them.
{"title":"Chirality Effects in Molecular Chainmail","authors":"Alexander R. Klotz, Caleb J. Anderson, Michael S. Dimitriyev","doi":"arxiv-2406.13590","DOIUrl":"https://doi.org/arxiv-2406.13590","url":null,"abstract":"Motivated by the observation of positive Gaussian curvature in kinetoplast\u0000DNA networks, we consider the effect of linking chirality in square lattice\u0000molecular chainmail networks using Langevin dynamics simulations and\u0000constrained gradient optimization. Linking chirality here refers to ordering of\u0000over-under versus under-over linkages between a loop and its neighbors. We\u0000consider fully alternating linking, maximally non-alternating, and partially\u0000non-alternating linking chiralities. We find that in simulations of polymer\u0000chainmail networks, the linking chirality dictates the sign of the Gaussian\u0000curvature of the final state of the chainmail membranes. Alternating networks\u0000have positive Gaussian curvature, similar to what is observed in kinetoplast\u0000DNA networks. Maximally non-alternating networks form isotropic membranes with\u0000negative Gaussian curvature. Partially non-alternating networks form flat\u0000diamond-shaped sheets which undergo a thermal folding transition when\u0000sufficiently large, similar to the crumpling transition in tethered membranes.\u0000We further investigate this topology-curvature relationship on geometric\u0000grounds by considering the tightest possible configurations and the constraints\u0000that must be satisfied to achieve them.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"140 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141524746","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We provide new techniques to construct sets of reals without perfect subsets and with the Hurewicz or Menger covering properties. In particular, we show that if the Continuum Hypothesis holds, then there are such sets which can be mapped continuously onto the Cantor space. These results allow to separate the properties of Menger and $mathsf{S}_1(Gamma,mathrm{O})$ in the realm of sets of reals without perfect subsets and solve a problem of Nowik and Tsaban concerning perfectly meager subsets in the transitive sense. We present also some other applications of the mentioned above methods.
{"title":"Small Hurewicz and Menger sets which have large continuous images","authors":"Piotr Szewczak, Tomasz Weiss, Lyubomyr Zdomskyy","doi":"arxiv-2406.12609","DOIUrl":"https://doi.org/arxiv-2406.12609","url":null,"abstract":"We provide new techniques to construct sets of reals without perfect subsets\u0000and with the Hurewicz or Menger covering properties. In particular, we show\u0000that if the Continuum Hypothesis holds, then there are such sets which can be\u0000mapped continuously onto the Cantor space. These results allow to separate the\u0000properties of Menger and $mathsf{S}_1(Gamma,mathrm{O})$ in the realm of sets\u0000of reals without perfect subsets and solve a problem of Nowik and Tsaban\u0000concerning perfectly meager subsets in the transitive sense. We present also\u0000some other applications of the mentioned above methods.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141524747","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In 2009, Caramello proved that each topos has a largest dense subtopos whose internal logic satisfies De Morgan law (also known as the law of the weak excluded middle). This finding implies that every locale has a largest dense extremally disconnected sublocale, referred to as its DeMorganization. In this paper, we take the first steps in exploring the DeMorganization in the localic context, shedding light on its geometric nature by showing that it is always a fitted sublocale and by providing a concrete description. The main result of the paper is that for any metrizable locale (without isolated points), its DeMorganization coincides with its Booleanization. This, in particular, implies that any extremally disconnected metric locale (without isolated points) must be Boolean, generalizing a well-known result for topological spaces to the localic setting.
{"title":"The DeMorganization of a locale","authors":"Igor Arrieta","doi":"arxiv-2406.12486","DOIUrl":"https://doi.org/arxiv-2406.12486","url":null,"abstract":"In 2009, Caramello proved that each topos has a largest dense subtopos whose\u0000internal logic satisfies De Morgan law (also known as the law of the weak\u0000excluded middle). This finding implies that every locale has a largest dense\u0000extremally disconnected sublocale, referred to as its DeMorganization. In this\u0000paper, we take the first steps in exploring the DeMorganization in the localic\u0000context, shedding light on its geometric nature by showing that it is always a\u0000fitted sublocale and by providing a concrete description. The main result of\u0000the paper is that for any metrizable locale (without isolated points), its\u0000DeMorganization coincides with its Booleanization. This, in particular, implies\u0000that any extremally disconnected metric locale (without isolated points) must\u0000be Boolean, generalizing a well-known result for topological spaces to the\u0000localic setting.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141524748","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the study of the Stone-u{C}ech remainder of the real line a detailed study of the Stone-u{C}ech remainder of the space $mathbb Ntimes [0,1]$, which we denote as $mathbb M$, has often been utilized. Of course the real line can be covered by two closed sets that are each homeomorphic to $mathbb M$. It is known that an autohomeomorphism of $mathbb M^*$ induces an autohomeomorphism of $mathbb N^*$. We prove that it is consistent with there being non-trivial autohomeomorphism of $mathbb N^*$ that those induced by autohomeomorphisms of $mathbb M^*$ are trivial.
{"title":"Autohomeomorphisms of pre-images of $mathbb N^*$","authors":"Alan Dow","doi":"arxiv-2406.09319","DOIUrl":"https://doi.org/arxiv-2406.09319","url":null,"abstract":"In the study of the Stone-u{C}ech remainder of the real line a detailed\u0000study of the Stone-u{C}ech remainder of the space $mathbb Ntimes [0,1]$,\u0000which we denote as $mathbb M$, has often been utilized. Of course the real\u0000line can be covered by two closed sets that are each homeomorphic to $mathbb\u0000M$. It is known that an autohomeomorphism of $mathbb M^*$ induces an\u0000autohomeomorphism of $mathbb N^*$. We prove that it is consistent with there\u0000being non-trivial autohomeomorphism of $mathbb N^*$ that those induced by\u0000autohomeomorphisms of $mathbb M^*$ are trivial.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"94 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141501280","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the article a technique of the usage of $f$-continuous functions (on mappings) and their families is developed. A proof of the Urysohn's Lemma for mappings is presented and a variant of the Brouwer-Tietze-Urysohn Extension Theorem for mappings is proven. Characterizations of the normality properties of mappings are given and the notion of a perfect normality of a mapping is introduced. It seems to be the most optimal in this approach.
{"title":"Functional approach to the normality of mappings","authors":"Mikhail Yourievich Liseev","doi":"arxiv-2406.08061","DOIUrl":"https://doi.org/arxiv-2406.08061","url":null,"abstract":"In the article a technique of the usage of $f$-continuous functions (on\u0000mappings) and their families is developed. A proof of the Urysohn's Lemma for\u0000mappings is presented and a variant of the Brouwer-Tietze-Urysohn Extension\u0000Theorem for mappings is proven. Characterizations of the normality properties\u0000of mappings are given and the notion of a perfect normality of a mapping is\u0000introduced. It seems to be the most optimal in this approach.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141532047","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For an index set $Gamma$ and a cardinal number $kappa$ the $Sigma_{kappa}$-product of real lines $Sigma_{kappa}(mathbb{R}^{Gamma})$ consist of all elements of $mathbb{R}^{Gamma}$ with $
{"title":"Function spaces on Corson-like compacta","authors":"Krzysztof Zakrzewski","doi":"arxiv-2406.07452","DOIUrl":"https://doi.org/arxiv-2406.07452","url":null,"abstract":"For an index set $Gamma$ and a cardinal number $kappa$ the\u0000$Sigma_{kappa}$-product of real lines $Sigma_{kappa}(mathbb{R}^{Gamma})$\u0000consist of all elements of $mathbb{R}^{Gamma}$ with $<kappa$ nonzero\u0000coordinates. A compact space is $kappa$-Corson if it can be embedded into\u0000$Sigma_{kappa}(mathbb{R}^{Gamma})$ for some $Gamma$. We also consider a\u0000class of compact spaces wider than the class of $omega$-Corson compact spaces,\u0000investigated by Nakhmanson and Yakovlev as well as Marciszewski, Plebanek and\u0000Zakrzewski called $NY$ compact spaces. For a Tychonoff space $X$, let\u0000$C_{p}(X)$ be the space of real continuous functions on the space $X$, endowed\u0000with the pointwise convergence topology. We present here a characterisation of\u0000$kappa$-Corson compact spaces $K$ for regular, uncountable cardinal numbers\u0000$kappa$ in terms of function spaces $C_{p}(K)$, extending a theorem of Bell\u0000and Marciszewski and a theorem of Pol. We also prove that classes of $NY$\u0000compact spaces and $omega$-Corson compact spaces $K$ are preserved by linear\u0000homeomorphisms of function spaces $C_{p}(K)$.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"64 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141501277","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $mathbb{R}^{+}=[0, infty)$ and let $mathbf{End}_{mathbb{R}^+}$ be the set of all endomorphisms of the monoid $(mathbb{R}^+, vee)$. The set $mathbf{End}_{mathbb{R}^+}$ is a monoid with respect to the operation of the function composition $g circ f$. It is shown that $g : mathbb{R}^+ to mathbb{R}^+$ is pseudometric-preserving iff $g in mathbf{End}_{mathbb{R}^+}$. In particular, a function $f : mathbb{R}^+ to mathbb{R}^+$ is ultrametric-preserving iff it is an endomorphism of $(mathbb{R}^+,vee)$ with kelnel consisting only the zero point. We prove that a given $mathbf{A} subseteq mathbf{End}_{mathbb{R}^+}$ is a submonoid of $(mathbf{End}, circ)$ iff there is a class $mathbf{X}$ of pseudoultrametric spaces such that $mathbf{A}$ coincides with the set of all functions which preserve the spaces from $mathbf{X}$. An explicit construction of such $mathbf{X}$ is given.
{"title":"Ultrametric-preserving functions as monoid endomorphisms","authors":"Oleksiy Dovgoshey","doi":"arxiv-2406.07166","DOIUrl":"https://doi.org/arxiv-2406.07166","url":null,"abstract":"Let $mathbb{R}^{+}=[0, infty)$ and let $mathbf{End}_{mathbb{R}^+}$ be the\u0000set of all endomorphisms of the monoid $(mathbb{R}^+, vee)$. The set\u0000$mathbf{End}_{mathbb{R}^+}$ is a monoid with respect to the operation of the\u0000function composition $g circ f$. It is shown that $g : mathbb{R}^+ to\u0000mathbb{R}^+$ is pseudometric-preserving iff $g in\u0000mathbf{End}_{mathbb{R}^+}$. In particular, a function $f : mathbb{R}^+ to\u0000mathbb{R}^+$ is ultrametric-preserving iff it is an endomorphism of\u0000$(mathbb{R}^+,vee)$ with kelnel consisting only the zero point. We prove that\u0000a given $mathbf{A} subseteq mathbf{End}_{mathbb{R}^+}$ is a submonoid of\u0000$(mathbf{End}, circ)$ iff there is a class $mathbf{X}$ of pseudoultrametric\u0000spaces such that $mathbf{A}$ coincides with the set of all functions which\u0000preserve the spaces from $mathbf{X}$. An explicit construction of such\u0000$mathbf{X}$ is given.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"57 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141524749","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Oleksandr Maslyuchenko, Vadym Myronyk, Roman Ivasiuk
We consider two natural topologies on the space $S(Xtimes Y,Z)$ of all separately continuous functions defined on the product of two topological spaces $X$ and $Y$ and ranged into a topological or metric space $X$. These topologies are the cross-open topology and the cross-uniform topology. We show that these topologies coincides if $X$ and $Y$ are pseudocompacts and $Z$ is a metric space. We prove that a compact space $K$ embeds into $S(Xtimes Y,Z)$ for infinite compacts $X$, $Y$ and a metrizable space $Zsupseteqmathbb{R}$ if and only if the weight of $K$ is less than the sharp cellularity of both spaces $X$ and $Y$.
{"title":"Compact subspaces of the space of separately continuous functions with the cross-uniform topology","authors":"Oleksandr Maslyuchenko, Vadym Myronyk, Roman Ivasiuk","doi":"arxiv-2406.05705","DOIUrl":"https://doi.org/arxiv-2406.05705","url":null,"abstract":"We consider two natural topologies on the space $S(Xtimes Y,Z)$ of all\u0000separately continuous functions defined on the product of two topological\u0000spaces $X$ and $Y$ and ranged into a topological or metric space $X$. These\u0000topologies are the cross-open topology and the cross-uniform topology. We show\u0000that these topologies coincides if $X$ and $Y$ are pseudocompacts and $Z$ is a\u0000metric space. We prove that a compact space $K$ embeds into $S(Xtimes Y,Z)$\u0000for infinite compacts $X$, $Y$ and a metrizable space $Zsupseteqmathbb{R}$ if\u0000and only if the weight of $K$ is less than the sharp cellularity of both spaces\u0000$X$ and $Y$.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141524750","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We investigate closed knight's tours on M"obius strip and Klein bottle chess boards. In particular, we characterize the board dimensions that admit tours that are nullhomotopic and the board dimensions that admit tours that realize generators of the fundamental groups of each of the surfaces.
{"title":"Nullhomotopic and Generating Knight's Tours on Non-Orientable Surfaces","authors":"Bradley Forrest, Zachary Lague","doi":"arxiv-2406.05226","DOIUrl":"https://doi.org/arxiv-2406.05226","url":null,"abstract":"We investigate closed knight's tours on M\"obius strip and Klein bottle chess\u0000boards. In particular, we characterize the board dimensions that admit tours\u0000that are nullhomotopic and the board dimensions that admit tours that realize\u0000generators of the fundamental groups of each of the surfaces.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"208 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141524751","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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