On generalized neighbourhood systems
Let \(X\) be a nonempty set. A generalized topology (GT) \(\mu\) on \(X\) is a subset of \(\exp X\) such that \(\emptyset \in \mu\) and every union of elements of \(\mu\) belongs to \(\mu\). A map \(\psi : X \rightarrow \exp X\) is called a generalized neighbourhood system (GNS) on \(X\) if, for \(x \in X\) and \(V\in \psi (x)\), we have \(x\in V\). Moreover, in this paper, two operations \(\iota_{\psi}\) and \(\gamma_{\psi}\) are defined: for \(A\subset X\), \(x\in \iota_{\psi}(A)\) iff there is \(V\in \psi (x)\) such that \(V\subset A\); \(x\in \gamma_{\psi}(A)\) iff \(V\in \psi (x)\) implies \(V\cap A\neq \emptyset\). In this paper, the author considers problems connected with generating a GT by a GNS and the operations \(\iota_{\psi}\) and \(\gamma_{\psi}\). The paper is finished by short considerations connected with \((\phi ,\psi )\)-continuity.
scientific article; zbMATH DE number 5651859