Recent zbMATH articles in MSC 54Dhttps://www.zbmath.org/atom/cc/54D2021-09-16T13:13:31.966056ZWerkzeugThe vertice-centered metric topologies generated from the connected undirected graphshttps://www.zbmath.org/1467.051442021-09-16T13:13:31.966056Z"Sari, H. K."https://www.zbmath.org/authors/?q=ai:sari.h-k"Kopuzlu, A."https://www.zbmath.org/authors/?q=ai:kopuzlu.abdullahSummary: Graph theory is used as a way of specifying relationships among a collection of items. As a result of this, the theory is important for many fields from science to liberal arts. Recently, topological structure of the graphs is an interesting research topic. In this paper, we study topological structures of the connected undirected graphs. We firstly define a metric on a connected undirected graph using the distance between vertices of the graph. We generate a vertice-centered metric topology on vertices set of a connected undirected graph using this metric. Moreover, we study some properties of this topology. Finally, the vertice-centered metric topologies on vertices set of certain graphs are studied.Solution of Ponomarev's problem of condensation onto compact setshttps://www.zbmath.org/1467.540022021-09-16T13:13:31.966056Z"Osipov, A. V."https://www.zbmath.org/authors/?q=ai:osipov.alexander-v"Pytkeev, E. G."https://www.zbmath.org/authors/?q=ai:pytkeev.evgenii-georgevichA condensation is a continuous bijection between topological spaces, or, if one identifies domain and image, passing to a smaller topology on the same set. The main question, due to Alexandroff, is when a Hausdorff space has a condensation to a compact Hausdorff space. A related problem is which compact Hausdorff spaces are \(a\)-spaces, which means that every co-countable subspace has a condensation to a compact Hausdorff space. Compact metric spaces are \(a\)-spaces, as are compact first-countable zero-dimensional and compact ordered spaces. Ponomarev asked whether perfectly normal compact spaces are \(a\)-spaces. In [\textit{E. G. Pytkeev}, Sov. Math., Dokl. 26, 162--165 (1982; Zbl 0529.54009); translation from Dokl. Akad. Nauk SSSR 265, 819--823 (1982)] the second author showed that first-countability (points are \(G_\delta\)) does not suffice. The present paper is devoted to the construction of a counterexample to Ponomarev's full question, under the assumption of the Continuum Hypothesis.Fréchet-like properties and almost disjoint familieshttps://www.zbmath.org/1467.540062021-09-16T13:13:31.966056Z"Corral, César"https://www.zbmath.org/authors/?q=ai:corral.cesar"Hrušák, Michael"https://www.zbmath.org/authors/?q=ai:hrusak.michaelThe authors study the relationship between \(\alpha_i\) properties and strong Fréchet-like properties in Isbell-Mrówka spaces. The main motivation is a question by G. Gruenhage: Is every \(\alpha_3\)-FU (hereditarily \(\alpha_3\)-FU) almost disjoint family bisequential?
To simplify the notation, they say that an almost disjoint family \(\mathcal A\) satisfies a topological property iff \(\omega\cup\{\infty\}\) (viewed as a subspace of the one-point compactification of the Isbell-Mrówka space \(\Psi(\mathcal A)\)) does, and that \(\mathcal A\) hereditarily satisfies a topological property iff for every \(\mathcal B\subseteq A\), \(\mathcal B\) satisfies this topological property in the previous sense.
In the second section of the paper, they prove that \(\text{non}(\mathcal M)=\mathfrak c\) implies that there exists an \(\alpha_3\)-FU almost disjoint family \(\mathcal A\) which is not hereditarily \(\alpha_3\)-FU, and that \(\mathfrak b=\mathfrak c\) implies that there is a hereditarily \(\alpha_3\)-FU almost disjoint family \(\mathcal A\) which is not bisequential.
In the third section, they get to the same conclusion under \(\mathfrak c\leq \aleph_2\) by proving that under \(\mathfrak s\leq \mathfrak b\), there is an \(\alpha_3\)-FU almost disjoint family \(\mathcal A\) which is not hereditarily \(\alpha_3\). The fourth section uses \(\Diamond(\mathfrak b)\) to prove this latter result.
In the last section, the authors sum up all the results and prove that under CH, there is a countable \(\alpha_1\) absolutely Fréchet space that is not bisequential.