The NZ property and \(D\)-spaces
type: Item
instance of: scholarly article
title: The NZ property and \(D\)-spaces
zbMATH Open document ID: 1389.54061
author: Ziqin Feng
author: Gary Gruenhage
published in: Acta Mathematica Hungarica
publication date: 2017-11-10T00:00:00Z
review text: On a topological space, if a function \(N\) is so defined that \(N(x)\) is an open neighborhood of the point \(x\) for each \(x\in X\), then \(N\) is called a neighborhood assignment on \(X\); if for any neighborhood assignment \(N\) of a space \(X\), there exists a closed and discrete subspace \(D\subset X\) with \(N(D)=\cup \{N(x): x \in D\}=X\), then \(X\) is called a \(D\)-space. \smallskip A function \(\mathcal{W}\) on \(X\) such that \(\mathcal{W}(x)\) is a collection of subsets of \(X\) containing \(x\) is called a point network for \(X\) and such a network satisfies \((F)\) if \newline \((F)\): whenever \(x\in U\) and \(U\) is open, there exists an open \(V=V(x,U)\) containing \(x\) such that \(y \in V\) implies \(x \in W \subset U\) for some \(W\in \mathcal{W}(y)\).\newline If each \(\mathcal{W}(x)\) is a chain of neighborhoods of \(x\) and satisfies \((F)\), then \(X\) is called a chain neighborhood \((F)\) space. \smallskip In the process of establishing a connection between chain neighborhood \((F)\) spaces and \(D\)-spaces, the concept of property \(NZ(\kappa)\) is brought in where \(\kappa\) is a cardinal; a topological space is said to satisfy this property if, for any collection \(\mathcal{N}=\{(x_{\lambda}, U_{\lambda}): \lambda \in \Lambda\}\), \(U_{\lambda}\) being an open neighborhood of \(x_{\lambda}\) for each \(\lambda \in \Lambda\), there exists a cover \(\{\Lambda_{\alpha}: \alpha <\kappa\}\) of \(\Lambda\) and an open set \(Z_{\lambda} \subset U_{\lambda}\) containing \(x_{\lambda}\), for each \(\lambda \in \Lambda\) such that for each \(\alpha <\kappa\): \newline \((\ast)\) For any sequence \(\{\lambda_{n}: n\in \omega\}\subset\Lambda_{\alpha}\), if \(x_{\lambda_{n}}\not\in \bigcup\limits_{i<n} U_{\lambda_{i}}\) for each \(n \in \omega\), then \(\bigcap\limits_{n\in \omega} Z_{\lambda_n}=\phi\).\newline It is shown that any space satisfying \(NZ(\omega)\) is a \(D\)-space; it is further shown that any chain neighborhood \((F)\) space has t…
Mathematics Subject Classification ID: 54D20
zbMATH DE Number: 6806065
zbMATH Keywords: NZ(\(\kappa\))-property
zbMATH Keywords: chain neighborhood \((F)\)-space
zbMATH Keywords: monotonically metacompact space
MaRDI profile type: Publication
full work available at URL: s10474-016-0587-0
OpenAlex ID: W2465318293
cites work: A Lattice of Conditions on Topological Spaces
cites work: Compact monotonically metacompact spaces are metrizable
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DOI: 10.1007/S10474-016-0587-0
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description: scientific article; zbMATH DE number 6806065