We define and characterize the notion of (i,j)-Baireness for bilocales. We also give internal properties of (i,j)-Baire bilocales which are not translated from properties of (i,j)-Baireness in bispaces. It turns out (i,j)-Baire bilocales are conservative in bilocales, in the sense that a bitopological space is almost (i,j)-Baire if and only if the bilocale it induces is (i,j)-Baire. Furthermore, in the class of Noetherian bilocales, (i,j)-Baireness of a bilocale coincides with (i,j)-Baireness of its ideal bilocale. We also consider relative versions of (i,j)-Baire where we show that a bilocale is (i,j)-Baire only if the subbilocale induced by the Booleanization is (i,j)-Baire. We use the characterization of (i,j)-Baire bilocales to introduce and characterize (tau_{i},tau_{j})-Baireness in the category of topobilocales.
{"title":"On the category of (i,j)-Baire Bilocales","authors":"Mbekezeli Nxumalo","doi":"arxiv-2407.13334","DOIUrl":"https://doi.org/arxiv-2407.13334","url":null,"abstract":"We define and characterize the notion of (i,j)-Baireness for bilocales. We\u0000also give internal properties of (i,j)-Baire bilocales which are not translated\u0000from properties of (i,j)-Baireness in bispaces. It turns out (i,j)-Baire\u0000bilocales are conservative in bilocales, in the sense that a bitopological\u0000space is almost (i,j)-Baire if and only if the bilocale it induces is\u0000(i,j)-Baire. Furthermore, in the class of Noetherian bilocales, (i,j)-Baireness\u0000of a bilocale coincides with (i,j)-Baireness of its ideal bilocale. We also\u0000consider relative versions of (i,j)-Baire where we show that a bilocale is\u0000(i,j)-Baire only if the subbilocale induced by the Booleanization is\u0000(i,j)-Baire. We use the characterization of (i,j)-Baire bilocales to introduce\u0000and characterize (tau_{i},tau_{j})-Baireness in the category of\u0000topobilocales.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141745478","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}
Given an open cover $mathcal{U}$ of a topological space $X$, we introduce the notion of a star network for $mathcal{U}$. The associated cardinal function $sn(X)$, where $e(X)leq sn(X)leq L(X)$, is used to establish new cardinal inequalities involving diagonal degrees. We show $|X|leq sn(X)^{Delta(X)}$ for a $T_1$ space $X$, giving a partial answer to a long-standing question of Angelo Bella. Many further results are given using variations of $sn(X)$. One result has as corollaries Buzyakova's theorem that a ccc space with a regular $G_delta$-diagonal has cardinality at most $mathfrak{c}$, as well as three results of Gotchev. Further results lead to logical improvements of theorems of Basile, Bella, and Ridderbos, a partial solution to a question of the same authors, and a theorem of Gotchev, Tkachenko, and Tkachuk. Finally, we define the Urysohn extent $Ue(X)$ with the property $Ue(X)leqmin{aL(X),e(X)}$ and use the ErdH{o}s-Rado theorem to show that $|X|leq 2^{Ue(X)overline{Delta}(X)}$ for any Urysohn space $X$.
{"title":"On diagonal degrees and star networks","authors":"Nathan Carlson","doi":"arxiv-2407.13508","DOIUrl":"https://doi.org/arxiv-2407.13508","url":null,"abstract":"Given an open cover $mathcal{U}$ of a topological space $X$, we introduce\u0000the notion of a star network for $mathcal{U}$. The associated cardinal\u0000function $sn(X)$, where $e(X)leq sn(X)leq L(X)$, is used to establish new\u0000cardinal inequalities involving diagonal degrees. We show $|X|leq\u0000sn(X)^{Delta(X)}$ for a $T_1$ space $X$, giving a partial answer to a\u0000long-standing question of Angelo Bella. Many further results are given using\u0000variations of $sn(X)$. One result has as corollaries Buzyakova's theorem that a\u0000ccc space with a regular $G_delta$-diagonal has cardinality at most\u0000$mathfrak{c}$, as well as three results of Gotchev. Further results lead to\u0000logical improvements of theorems of Basile, Bella, and Ridderbos, a partial\u0000solution to a question of the same authors, and a theorem of Gotchev,\u0000Tkachenko, and Tkachuk. Finally, we define the Urysohn extent $Ue(X)$ with the\u0000property $Ue(X)leqmin{aL(X),e(X)}$ and use the ErdH{o}s-Rado theorem to\u0000show that $|X|leq 2^{Ue(X)overline{Delta}(X)}$ for any Urysohn space $X$.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141737199","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 discuss conditions under which certain compactifications of topological spaces can be obtained by composing the ultrafilter space monad with suitable reflectors. In particular, we show that these compactifications inherit their categorical properties from the ultrafilter space monad. We further observe that various constructions such as the prime open filter monad defined by H. Simmons, the prime closed filter compactification studied by Bentley and Herrlich, as well as the separated completion monad studied by Salbany fall within the same categorical framework.
{"title":"Separated and prime compactifications","authors":"Ando Razafindrakoto","doi":"arxiv-2407.11538","DOIUrl":"https://doi.org/arxiv-2407.11538","url":null,"abstract":"We discuss conditions under which certain compactifications of topological\u0000spaces can be obtained by composing the ultrafilter space monad with suitable\u0000reflectors. In particular, we show that these compactifications inherit their\u0000categorical properties from the ultrafilter space monad. We further observe\u0000that various constructions such as the prime open filter monad defined by H.\u0000Simmons, the prime closed filter compactification studied by Bentley and\u0000Herrlich, as well as the separated completion monad studied by Salbany fall\u0000within the same categorical framework.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"38 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141720038","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}
This is a survey of recent and classical results concerning various types of homogeneity, such as n-homogeneity, discrete homogeneity, and countable dense homogeneity. Some new results are also presented, and several problems are posed.
{"title":"From homogeneity to discrete homogeneity","authors":"Vitalij A. Chatyrko, Alexandre Karassev","doi":"arxiv-2407.11815","DOIUrl":"https://doi.org/arxiv-2407.11815","url":null,"abstract":"This is a survey of recent and classical results concerning various types of\u0000homogeneity, such as n-homogeneity, discrete homogeneity, and countable dense\u0000homogeneity. Some new results are also presented, and several problems are\u0000posed.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"72 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141720037","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}
Given an ideal $mathcal{I}$ on the nonnegative integers $omega$ and a Polish space $X$, let $mathscr{L}(mathcal{I})$ be the family of subsets $Ssubseteq X$ such that $S$ is the set of $mathcal{I}$-limit points of some sequence taking values in $X$. First, we show that $mathscr{L}(mathcal{I})$ may attain arbitrarily large Borel complexity. Second, we prove that if $mathcal{I}$ is a $G_{deltasigma}$-ideal then all elements of $mathscr{L}(mathcal{I})$ are closed. Third, we show that if $mathcal{I}$ is a simply coanalytic ideal and $X$ is first countable, then every element of $mathscr{L}(mathcal{I})$ is simply analytic. Lastly, we studied certain structural properties and the topological complexity of minimal ideals $mathcal{I}$ for which $mathscr{L}(mathcal{I})$ contains a given set.
{"title":"Topological complexity of ideal limit points","authors":"Marek Balcerzak, Szymon Glab, Paolo Leonetti","doi":"arxiv-2407.12160","DOIUrl":"https://doi.org/arxiv-2407.12160","url":null,"abstract":"Given an ideal $mathcal{I}$ on the nonnegative integers $omega$ and a\u0000Polish space $X$, let $mathscr{L}(mathcal{I})$ be the family of subsets\u0000$Ssubseteq X$ such that $S$ is the set of $mathcal{I}$-limit points of some\u0000sequence taking values in $X$. First, we show that $mathscr{L}(mathcal{I})$\u0000may attain arbitrarily large Borel complexity. Second, we prove that if\u0000$mathcal{I}$ is a $G_{deltasigma}$-ideal then all elements of\u0000$mathscr{L}(mathcal{I})$ are closed. Third, we show that if $mathcal{I}$ is\u0000a simply coanalytic ideal and $X$ is first countable, then every element of\u0000$mathscr{L}(mathcal{I})$ is simply analytic. Lastly, we studied certain\u0000structural properties and the topological complexity of minimal ideals\u0000$mathcal{I}$ for which $mathscr{L}(mathcal{I})$ contains a given set.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"48 15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141737200","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 examine conditions under which projective limits of topological spaces are preserved by the continuous valuation functor $mathbf V$ and its subprobability and probability variants (used to represent probabilistic choice), by the Smyth hyperspace functor (demonic non-deterministic choice), by the Hoare hyperspace functor (angelic non-deterministic choice), by Heckmann's $mathbf A$-valuation functor, by the quasi-lens functor, by the Plotkin hyperspace functor (erratic non-deterministic choice), and by prevision functors and powercone functors that implement mixtures of probabilistic and non-deterministic choice.
{"title":"On the Preservation of Projective Limits by Functors of Non-Deterministic, Probabilistic, and Mixed Choice","authors":"Jean Goubault-Larrecq","doi":"arxiv-2407.10235","DOIUrl":"https://doi.org/arxiv-2407.10235","url":null,"abstract":"We examine conditions under which projective limits of topological spaces are\u0000preserved by the continuous valuation functor $mathbf V$ and its\u0000subprobability and probability variants (used to represent probabilistic\u0000choice), by the Smyth hyperspace functor (demonic non-deterministic choice), by\u0000the Hoare hyperspace functor (angelic non-deterministic choice), by Heckmann's\u0000$mathbf A$-valuation functor, by the quasi-lens functor, by the Plotkin\u0000hyperspace functor (erratic non-deterministic choice), and by prevision\u0000functors and powercone functors that implement mixtures of probabilistic and\u0000non-deterministic choice.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141720039","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 show that if $X$ is an arc-like continuum, then for every point $x in X$ there is a plane embedding of $X$ in which $x$ is an accessible point. This answers a question posed by Nadler in 1972, which has become known as the Nadler-Quinn problem in continuum theory. Towards this end, we develop the theories of truncations and contour factorizations of interval maps. As a corollary, we answer a question of Mayer from 1982 about inequivalent plane embeddings of indecomposable arc-like continua.
{"title":"The Nadler-Quinn problem on accessible points of arc-like continua","authors":"Andrea Ammerlaan, Ana Anušić, Logan C. Hoehn","doi":"arxiv-2407.09677","DOIUrl":"https://doi.org/arxiv-2407.09677","url":null,"abstract":"We show that if $X$ is an arc-like continuum, then for every point $x in X$\u0000there is a plane embedding of $X$ in which $x$ is an accessible point. This\u0000answers a question posed by Nadler in 1972, which has become known as the\u0000Nadler-Quinn problem in continuum theory. Towards this end, we develop the\u0000theories of truncations and contour factorizations of interval maps. As a\u0000corollary, we answer a question of Mayer from 1982 about inequivalent plane\u0000embeddings of indecomposable arc-like continua.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141720040","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 consider properties of the diagonal of a continuum that are used later in the paper. We continue the study of $T$-closed subsets of a continuum $X$. We prove that for a continuum $X$, the statements: $Delta_X$ is a nonblock subcontinuum of $X^2$, $Delta_X$ is a shore subcontinuum of $X^2$ and $Delta_X$ is not a strong centre of $X^2$ are equivalent, this result answers in the negative Questions 35 and 36 and Question 38 ($iin{4,5}$) of the paper ``Diagonals on the edge of the square of a continuum, by A. Illanes, V. Mart'inez-de-la-Vega, J. M. Mart'inez-Montejano and D. Michalik''. We also include an example, giving a negative answer to Question 1.2 of the paper ``Concerning when $F_1(X)$ is a continuum of colocal connectedness in hyperspaces and symmetric products, Colloquium Math., 160 (2020), 297-307'', by V. Mart'inez-de-la-Vega, J. M. Mart'inez-Montejano. We characterised the $T$-closed subcontinua of the square of the pseudo-arc. We prove that the $T$-closed sets of the product of two continua is compact if and only if such product is locally connected. We show that for a chainable continuum $X$, $Delta_X$ is a $T$-closed subcontinuum of $X^2$ if and only if $X$ is an arc. We prove that if $X$ is a continuum with the property of Kelley, then the following are equivalent: $Delta_X$ is a $T$-closed subcontinuum of $X^2$, $X^2setminusDelta_X$ is strongly continuumwise connected, $Delta_X$ is a subcontinuum of colocal connectedness, and $X^2setminusDelta_X$ is continuumwise connected. We give models for the families of $T$-closed sets and $T$-closed subcontinua of various families of continua.
我们将考虑本文后面将用到的连续体对角线的性质。我们继续研究连续统 $X$ 的 $T$ 封闭子集。我们证明，对于连续统 $X$，以下陈述：这个结果回答了论文 "连续体正方形边缘上的对角线 "中的问题 35 和 36 以及问题 38 ($iin{4,5}$)。Illanes, V.Mart'inez-de-la-Vega, J. M. Mart'inez-Montejano and D. Michalik''。我们还包括一个例子，给出了论文 "Concerning when $F_1(X)$ is a continuum of colocal connectedness inhyperspaces and symmetric products, Colloquium Math., 160 (2020), 297-307'' 中问题 1.2 的否定答案，作者 V. Mart'inez-de-la-Vega.Mart'inez-de-la-Vega, J. M. Mart'inez-Montejano.我们描述了伪弧正方形的$T$封闭子洲的特征。我们证明，当且仅当两个连续体的乘积是局部连通时，该乘积的$T$闭集才是紧凑的。我们证明，对于可链连续统 $X$，当且仅当 $X$ 是弧时，$Delta_X$ 是 $X^2$ 的 $T$ 闭子连续统。我们证明，如果 $X$ 是一个具有 Kelley 特性的连续统，那么以下情况是等价的：$Delta_X$ 是 $X^2$ 的一个 $T$ 闭合的子连续统，$X^2setminusDelta_X$ 是强连续统连接，$Delta_X$ 是局部连接的子连续统，并且 $X^2setminusDelta_X$ 是连续统连接。我们给出了 $T$ 闭集族和各种连续体族的 $T$ 闭子连续体的模型。
{"title":"More on $mathcal{T}$-closed sets","authors":"Javier Camargo, Sergio Macías","doi":"arxiv-2407.09258","DOIUrl":"https://doi.org/arxiv-2407.09258","url":null,"abstract":"We consider properties of the diagonal of a continuum that are used later in\u0000the paper. We continue the study of $T$-closed subsets of a continuum $X$. We\u0000prove that for a continuum $X$, the statements: $Delta_X$ is a nonblock\u0000subcontinuum of $X^2$, $Delta_X$ is a shore subcontinuum of $X^2$ and\u0000$Delta_X$ is not a strong centre of $X^2$ are equivalent, this result answers\u0000in the negative Questions 35 and 36 and Question 38 ($iin{4,5}$) of the\u0000paper ``Diagonals on the edge of the square of a continuum, by A. Illanes, V.\u0000Mart'inez-de-la-Vega, J. M. Mart'inez-Montejano and D. Michalik''. We also\u0000include an example, giving a negative answer to Question 1.2 of the paper\u0000``Concerning when $F_1(X)$ is a continuum of colocal connectedness in\u0000hyperspaces and symmetric products, Colloquium Math., 160 (2020), 297-307'', by\u0000V. Mart'inez-de-la-Vega, J. M. Mart'inez-Montejano. We characterised the\u0000$T$-closed subcontinua of the square of the pseudo-arc. We prove that the\u0000$T$-closed sets of the product of two continua is compact if and only if such\u0000product is locally connected. We show that for a chainable continuum $X$,\u0000$Delta_X$ is a $T$-closed subcontinuum of $X^2$ if and only if $X$ is an arc.\u0000We prove that if $X$ is a continuum with the property of Kelley, then the\u0000following are equivalent: $Delta_X$ is a $T$-closed subcontinuum of $X^2$,\u0000$X^2setminusDelta_X$ is strongly continuumwise connected, $Delta_X$ is a\u0000subcontinuum of colocal connectedness, and $X^2setminusDelta_X$ is\u0000continuumwise connected. We give models for the families of $T$-closed sets and\u0000$T$-closed subcontinua of various families of continua.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141720041","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 prove that a compact space $K$ embeds into a $sigma$-product of compact metrizable spaces ($sigma$-product of intervals) if and only if $K$ is (strongly countable-dimensional) hereditarily metalindel"of and every subspace of $K$ has a nonempty relative open second-countable subset. This provides novel characterizations of $omega$-Corson and $NY$ compact spaces. We give an example of a uniform Eberlein compact space that does not embed into a product of compact metric spaces in such a way that the $sigma$-product is dense in the image. In particular, this answers a question of Kubi's and Leiderman. We also show that for a compact space $K$ the property of being $NY$ compact is determined by the topological structure of the space $C_p(K)$ of continuous real-valued functions of $K$ equipped with the pointwise convergence topology. This refines a recent result of Zakrzewski.
我们证明，当且仅当$K$是（强可数维的）遗传金属indel "的，并且$K$的每个子空间都有一个非空的相对开放的第二可数子集时，紧凑空间$K$嵌入到紧凑可三维空间的$sigma$-product（$sigma$-product of intervals）中。这提供了$omega$-Corson和$NY$紧凑空间的新特征。我们给出了一个均匀埃伯林紧凑空间的例子，它不会以这样一种方式嵌入到紧凑度量空间的乘积中，即$sigma$乘积在图像中是致密的。这尤其回答了库比（Kubi's）和莱德曼（Leiderman）的一个问题。我们还证明，对于一个紧凑空间 $K$ 而言，$NY$ 紧凑的性质是由装有点收敛拓扑的连续实值函数空间 $C_p(K)$ 的拓扑结构决定的。
{"title":"On the class of NY compact spaces of finitely supported elements and related classes","authors":"Antonio Avilés, Mikołaj Krupski","doi":"arxiv-2407.09090","DOIUrl":"https://doi.org/arxiv-2407.09090","url":null,"abstract":"We prove that a compact space $K$ embeds into a $sigma$-product of compact\u0000metrizable spaces ($sigma$-product of intervals) if and only if $K$ is\u0000(strongly countable-dimensional) hereditarily metalindel\"of and every subspace\u0000of $K$ has a nonempty relative open second-countable subset. This provides\u0000novel characterizations of $omega$-Corson and $NY$ compact spaces. We give an\u0000example of a uniform Eberlein compact space that does not embed into a product\u0000of compact metric spaces in such a way that the $sigma$-product is dense in\u0000the image. In particular, this answers a question of Kubi's and Leiderman. We\u0000also show that for a compact space $K$ the property of being $NY$ compact is\u0000determined by the topological structure of the space $C_p(K)$ of continuous\u0000real-valued functions of $K$ equipped with the pointwise convergence topology.\u0000This refines a recent result of Zakrzewski.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"324 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141720042","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}
Erin Wolf Chambers, Elizabeth Munch, Sarah Percival, Xinyi Wang
Geometric graphs appear in many real-world data sets, such as road networks, sensor networks, and molecules. We investigate the notion of distance between embedded graphs and present a metric to measure the distance between two geometric graphs via merge trees. In order to preserve as much useful information as possible from the original data, we introduce a way of rotating the sublevel set to obtain the merge trees via the idea of the directional transform. We represent the merge trees using a surjective multi-labeling scheme and then compute the distance between two representative matrices. We show some theoretically desirable qualities and present two methods of computation: approximation via sampling and exact distance using a kinetic data structure, both in polynomial time. We illustrate its utility by implementing it on two data sets.
{"title":"A Distance for Geometric Graphs via the Labeled Merge Tree Interleaving Distance","authors":"Erin Wolf Chambers, Elizabeth Munch, Sarah Percival, Xinyi Wang","doi":"arxiv-2407.09442","DOIUrl":"https://doi.org/arxiv-2407.09442","url":null,"abstract":"Geometric graphs appear in many real-world data sets, such as road networks,\u0000sensor networks, and molecules. We investigate the notion of distance between\u0000embedded graphs and present a metric to measure the distance between two\u0000geometric graphs via merge trees. In order to preserve as much useful\u0000information as possible from the original data, we introduce a way of rotating\u0000the sublevel set to obtain the merge trees via the idea of the directional\u0000transform. We represent the merge trees using a surjective multi-labeling\u0000scheme and then compute the distance between two representative matrices. We\u0000show some theoretically desirable qualities and present two methods of\u0000computation: approximation via sampling and exact distance using a kinetic data\u0000structure, both in polynomial time. We illustrate its utility by implementing\u0000it on two data sets.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"51 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141719999","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：