{"title":"On free bases of Banach spaces","authors":"E. Pernecká, J. Spěvák","doi":"arxiv-2405.03556","DOIUrl":"https://doi.org/arxiv-2405.03556","url":null,"abstract":"We call a closed subset M of a Banach space X a free basis of X if it\u0000contains the null vector and every Lipschitz map from M to a Banach space Y,\u0000which preserves the null vectors can be uniquely extended to a bounded linear\u0000map from X to Y. We then say that two complete metric spaces M and N are\u0000Mol-equivalent if they admit bi-Lipschitz copies M' and N', respectively that\u0000are free bases of a common Banach space satisfying span M'=span N'. In this note, we compare Mol-equivalence with some other natural equivalences\u0000on the class of complete metric spaces. The main result states that\u0000Mol-equivalent spaces must have the same v{C}ech-Lebesgue covering dimension.\u0000In combination with the work of Godard, this implies that two complete metric\u0000spaces with isomorphic Lipschitz-free spaces need not be Mol-equivalent. Also,\u0000there exist non-homeomorphic Mol-equivalent metric spaces, and, in contrast\u0000with the covering dimension, the metric Assouad dimension is not preserved by\u0000Mol-equivalence.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140883196","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}
{"title":"On $n$-dimensional Niemytzki spaces","authors":"Vitalij A. Chatyrko","doi":"arxiv-2405.02708","DOIUrl":"https://doi.org/arxiv-2405.02708","url":null,"abstract":"In this paper we extend the construction of the Niemytzki plane to dimension\u0000$n geq 3$ and explore some properties of the new spaces. Furthermore, we\u0000consider a poset of topologies on the closed $n$-dimensional Euclidean\u0000half-space similar to one from cite{AAK} which is related to the Niemytzki\u0000plane topology.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"62 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140883092","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}
Judyta Bąk, Taras Banakh, Joanna Garbulińska-Węgrzyn, Magdalena Nowak, Michał Popławski
{"title":"Characterizing Lipschitz images of injective metric spaces","authors":"Judyta Bąk, Taras Banakh, Joanna Garbulińska-Węgrzyn, Magdalena Nowak, Michał Popławski","doi":"arxiv-2405.01860","DOIUrl":"https://doi.org/arxiv-2405.01860","url":null,"abstract":"A metric space $X$ is {em injective} if every non-expanding map $f:Bto X$\u0000defined on a subspace $B$ of a metric space $A$ can be extended to a\u0000non-expanding map $bar f:Ato X$. We prove that a metric space $X$ is a\u0000Lipschitz image of an injective metric space if and only if $X$ is Lipschitz\u0000connected in the sense that for every points $x,yin X$, there exists a\u0000Lipschitz map $f:[0,1]to X$ such that $f(0)=x$ and $f(1)=y$. In this case the\u0000metric space $X$ carries a well-defined intrinsic metric. A metric space $X$ is\u0000a Lipschitz image of a compact injective metric space if and only if $X$ is\u0000compact, Lipschitz connected and its intrinsic metric is totally bounded. A\u0000metric space $X$ is a Lipschitz image of a separable injective metric space if\u0000and only if $X$ is a Lipschitz image of the Urysohn universal metric space if\u0000and only if $X$ is analytic, Lipschitz connected and its intrinsic metric is\u0000separable.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140883223","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}
{"title":"Endpoints of smooth plane dendroids","authors":"David S. Lipham","doi":"arxiv-2405.01706","DOIUrl":"https://doi.org/arxiv-2405.01706","url":null,"abstract":"We show that each endpoint of a smooth plane dendroid $X$ is accessible, and\u0000that the endpoint set $E(X)$ is circle-like in that every two of its points are\u0000separated by two other points. Also if $E(X)$ is totally disconnected and\u0000$1$-dimensional, then $X$ must contain an uncountable collection of\u0000pairwise-disjoint arcs. An example is constructed to show that this is false\u0000outside the plane.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"47 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140889886","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}
Alan Dow, Klaas Pieter Hart, Jan van Mill, Hans Vermeer
{"title":"$C$-embedding, Lindelöfness, Čech-completeness","authors":"Alan Dow, Klaas Pieter Hart, Jan van Mill, Hans Vermeer","doi":"arxiv-2404.19703","DOIUrl":"https://doi.org/arxiv-2404.19703","url":null,"abstract":"We show that in the class of Lindel\"of v{C}ech-complete spaces the property\u0000of being $C$-embedded is quite well-behaved. It admits a useful\u0000characterization that can be used to show that products and perfect preimages\u0000of $C$-embedded spaces are again $C$-embedded. We also show that both\u0000properties, Lindel\"of and v{C}ech-complete, are needed in the product result.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140840538","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}
{"title":"A Few Projective Classes of (Non-Hausdorff) Topological Spaces","authors":"Jean Goubault-Larrecq","doi":"arxiv-2404.18614","DOIUrl":"https://doi.org/arxiv-2404.18614","url":null,"abstract":"A class of topological spaces is projective (resp., $omega$-projective) if\u0000and only if projective systems of spaces (resp., with a countable cofinal\u0000subset of indices) in the class are still in the class. A certain number of\u0000classes of Hausdorff spaces are known to be, or not to be, ($omega$-)\u0000projective. We examine classes of spaces that are not necessarily Hausdorff.\u0000Sober and compact sober spaces form projective classes, but most classes of\u0000locally compact spaces are not even $omega$-projective. Guided by the fact\u0000that the stably compact spaces are exactly the locally compact, strongly sober\u0000spaces, and that the strongly sober spaces are exactly the sober, coherent,\u0000compact, weakly Hausdorff (in the sense of Keimel and Lawson) spaces, we\u0000examine which classes defined by combinations of those properties are\u0000projective. Notably, we find that coherent sober spaces, compact coherent sober\u0000spaces, as well as (locally) strongly sober spaces, form projective classes.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140840545","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}
{"title":"Canonical extensions via fitted sublocales","authors":"Tomáš Jakl, Anna Laura Suarez","doi":"arxiv-2404.18325","DOIUrl":"https://doi.org/arxiv-2404.18325","url":null,"abstract":"We build on a recent result stating that the frame $mathsf{SE}(L)$ of\u0000strongly exact filters for a frame $L$ is anti-isomorphic to the coframe\u0000$mathsf{S}_o(L)$ of fitted sublocales. The collection $mathsf{E}(L)$ of exact\u0000filters of $L$ is known to be a sublocale of this frame. We consider several\u0000other subcollections of $mathsf{SE}(L)$: the collections\u0000$mathcal{J}(mathsf{CP}(L))$ and $mathcal{J}(mathsf{SO}(L))$ of\u0000intersections of completely prime and Scott-open filters, respectively, and the\u0000collection $mathsf{R}(L)$ of regular elements of the frame of filters. We show\u0000that all of these are sublocales of $mathsf{SE}(L)$, and as such they\u0000correspond to subcolocales of $mathsf{S}_o(L)$, which all turn out to have a\u0000concise description. By using the theory of polarities of Birkhoff, one can\u0000show that all of the structures mentioned above enjoy universal properties\u0000which are variations of that of the canonical extension. We also show how some\u0000of these subcollections can be described as polarities and give three new\u0000equivalent definitions of subfitness in terms of the lattice of filters.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140840637","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}
{"title":"Sufficiently many projections in archimedean vector lattices with weak order unit","authors":"Anthony W. Hager, Brian Wynne","doi":"arxiv-2404.17628","DOIUrl":"https://doi.org/arxiv-2404.17628","url":null,"abstract":"The property of a vector lattice of sufficiently many projections (SMP) is\u0000informed by restricting attention to archimedean $A$ with a distinguished weak\u0000order unit $u$ (the class, or category, $bf{W}$), where the Yosida\u0000representation $A leq D(Y(A,u))$ is available. Here, $A$ SMP is equivalent to\u0000$Y(A,u)$ having a $pi$-base of clopen sets of a certain type called ``local\".\u0000If the unit is strong, all clopen sets are local and $A$ is SMP if and only if\u0000$Y(A,u)$ has clopen $pi$-base, a property we call $pi$-zero-dimensional\u0000($pi$ZD). The paper is in two parts: the first explicates the similarities of\u0000SMP and $pi$ZD; the second consists of examples, including $pi$ZD but not\u0000SMP, and constructions of many SMP's which seem scarce in the literature.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"45 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140840638","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}
Gaogao Dong, Nannan Sun, Fan Wang, Renaud Lambiotte
{"title":"Network shell structure based on hub and non-hub nodes","authors":"Gaogao Dong, Nannan Sun, Fan Wang, Renaud Lambiotte","doi":"arxiv-2404.17231","DOIUrl":"https://doi.org/arxiv-2404.17231","url":null,"abstract":"The shell structure holds significant importance in various domains such as\u0000information dissemination, supply chain management, and transportation. This\u0000study focuses on investigating the shell structure of hub and non-hub nodes,\u0000which play important roles in these domains. Our framework explores the\u0000topology of Erd\"{o}s-R'{e}nyi (ER) and Scale-Free (SF) networks, considering\u0000source node selection strategies dependent on the nodes' degrees. We define the\u0000shell $l$ in a network as the set of nodes at a distance $l$ from a given node\u0000and represent $r_l$ as the fraction of nodes outside shell $l$. Statistical\u0000properties of the shells are examined for a selected node, taking into account\u0000the node's degree. For a network with a given degree distribution, we\u0000analytically derive the degree distribution and average degree of nodes outside\u0000shell $l$ as functions of $r_l$. Moreover, we discover that $r_l$ follows an\u0000iterative functional form $r_l = phi(r_{l-1})$, where $phi$ is expressed in\u0000terms of the generating function of the original degree distribution of the\u0000network.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"73 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140809328","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}
Leandro Fiorini Aurichi, Paulo Magalhães Júnior, Lucas Real
{"title":"Topological remarks on end and edge-end spaces","authors":"Leandro Fiorini Aurichi, Paulo Magalhães Júnior, Lucas Real","doi":"arxiv-2404.17116","DOIUrl":"https://doi.org/arxiv-2404.17116","url":null,"abstract":"The notion of ends in an infinite graph $G$ might be modified if we consider\u0000them as equivalence classes of infinitely edge-connected rays, rather than\u0000equivalence classes of infinitely (vertex-)connected ones. This alternative\u0000definition yields to the edge-end space $Omega_E(G)$ of $G$, in which we can\u0000endow a natural (edge-)end topology. For every graph $G$, this paper proves\u0000that $Omega_E(G)$ is homeomorphic to $Omega(H)$ for some possibly another\u0000graph $H$, where $Omega(H)$ denotes its usual end space. However, we also show\u0000that the converse statement does not hold: there is a graph $H$ such that\u0000$Omega(H)$ is not homeomorphic to $Omega_E(G)$ for any other graph $G$. In\u0000other words, as a main result, we conclude that the class of topological spaces\u0000$Omega_E = {Omega_E(G) : G text{ graph}}$ is strictly contained in $Omega\u0000= {Omega(H) : H text{ graph}}$.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140809346","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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