{"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}
{"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}
{"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}
{"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}
{"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}
{"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}
{"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}
{"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}
{"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
{"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}
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