On certain generalized versions of groupability
type: Item
instance of: scholarly article
title: On certain generalized versions of groupability
zbMATH Open document ID: 1420.54035
author: Pratulananda Das
author: Upasana Samanta
author: Debraj Chandra
published in: Topology and its Applications
publication date: 2019-04-24T00:00:00Z
review text: Selection principles in topology have been widely studied in the literature. In this paper, the authors study selection principles in terms of stars, ideals and groupability. Recall that for a subset $A$ of a space $X$ and a family ${\mathcal P}$ of subsets of $X$, the \textit{star of} $A$ with respect to ${\mathcal P}$, denoted by $St(A,{\mathcal P})$, is the set $\bigcup\{P\in{\mathcal P}:A\cap P\neq\emptyset\}$; a family ${\mathcal I}$ of subsets of $X$ is said to be an \textit{ideal} of $X$ if for any $A,B\in{\mathcal I}$, we have that $A\cup B\in{\mathcal I}$, and if $A\in{\mathcal I}$ and $B\subset A$, then $B\in{\mathcal I}$; an element $A$ of a family ${\mathcal A}$ is said to be \textit{groupable} if there is a partition $A=\bigcup_{n\in\mathbb N} A_n$ of $A$ into pairwise disjoint finite sets such that for each infinite subset $N$ of the set of all positive integers, $\bigcup\{A_n:n\in N\}\in{\mathcal A}$, cf. [\textit{Lj. D. R. Kočinac} and \textit{M. Scheepers}, Fundam. Math. 179, No. 2, 131--155 (2003; Zbl 1115.91013)]. Two sample results: \textbf{Theorem 3.2.} For a paracompact space \(X\) if TWO has a winning strategy in the \(S\mathcal{I}H\) game on \(X\) then each \(\mathcal{I}\)-large cover is a star-\(\mathcal{I}\)-groupable as well as an \(\mathcal{I}\)-groupable open cover of \(X\). \textbf{Theorem 4.1.} For a space \(X\) \begin{itemize} \item[(i)] \(SS^*_{fin}(\mathcal{O}, \mathcal{R})\) and \(SS^*_{fin}(\mathcal{O}, \mathcal{O}^{\mathcal{R}-gp})\) are equivalent. \item[(ii)] \(U^*_{fin}(\mathcal{O}, \mathcal{R})\) and \(U^*_{fin}(\mathcal{O}, \mathcal{O}^{\mathcal{R}-gp})\) are equivalent. \item[(iii)] \(^*U_{fin}(\mathcal{O}, \mathcal{R})\) and \(^*U_{fin}(\mathcal{O}, \mathcal{O}^{\mathcal{R}-gp})\) are equivalent. \end{itemize}
reviewed by: Fortunata Aurora Basile
Mathematics Subject Classification ID: 54D20
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 91A44
zbMATH DE Number: 7047507
zbMATH Keywords: ideal
zbMATH Keywords: Hurewicz property
zbMATH Keywords: \(\mathcal{I}\)-\(\gamma\)-cover
zbMATH Keywords: \(\mathcal{I}\)-large cover
zbMATH Keywords: \(\mathcal{R}\)-groupability
zbMATH Keywords: star-\(\mathcal{I}\)-groupability
MaRDI profile type: Publication
full work available at URL: j.topol.2019.02.052
OpenAlex ID: W2918663025
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DOI: 10.1016/J.TOPOL.2019.02.052
description: scientific article; zbMATH DE number 7047507