The NZ property and \(D\)-spaces
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 the \(NZ\) property. \smallskip \,\,\,\, In the process of studying the \(NZ(\kappa)\) property, it is shown that it is closed under arbitrary subspaces and finite products and also for countable products if \(\kappa\) is infinite. \smallskip In an attempt to answer the long-standing question whether chain neighborhood \((F)\) spaces are \(D\)-spaces, it is shown that any countable product of chain neighborhood \((F)\) spaces is a hereditary \(D\)-space.
scientific article; zbMATH DE number 6806065
scientific article