On \(\mathcal{I}\)-neighborhood spaces and \(\mathcal{I}\)-quotient spaces
In this paper the author studies properties of ideal topological spaces, based on the new notion of \(\mathcal{I}_{sn}\)-open sets and related meanings such as the \(\mathcal{I}_{sn}\)-closure and \(\mathcal{I}_{sn}\)-interior operators and \(\mathcal{I}_{sn}\)-continuous mappings. Also, the author defines the so-called \(\mathcal{I}\)-neighborhood spaces, the \(\mathcal{I}_{sn}\)-open topology, denoted by \(\tau_{\mathcal{I}_{sn}}\), and \(\mathcal{I}_{sn}\)-open topological spaces, denoted by \(X_{\mathcal{I}_{sn}}\). Some of the main results of the paper are stated as follows: \begin{itemize} \item[(1)] A topological space \((X,\tau)\) is an \(\mathcal{I}\)-sequential space if and only if \(X_{\mathcal{I}_{sn}}\) is an \(\mathcal{I}\)-sequential space and \(\tau=\tau_{\mathcal{I}_{sn}}\). \item[(2)] A mapping \(f:X\rightarrow Y\) preserves \(\mathcal{I}\)-convergence if and only if provided \(U\) is an \(\mathcal{I}_{sn}\)-open subset of \(Y\), then \(f^{-1}(U)\) is an \(\mathcal{I}_{sn}\)-open subset of \(X\). \item[(3)] A topological space \(X\) is an \(\mathcal{I}\)-neighborhood space if and only if every \(\mathcal{I}\)-continuous mapping on the space \(X\) is an \(\mathcal{I}_{sn}\)-continuous mapping. \end{itemize} Moreover, in order to characterize the finest topology of the range that makes the mapping from a topological space onto a set preserve \(\mathcal{I}\)-convergence (be \(\mathcal{I}_{sn}\)-continuous), the author introduces \(\mathcal{I}_{sn}\)-quotient mappings and discusses some topological properties of \(\mathcal{I}_{sn}\)-quotient spaces. Continuing with the essential results of the paper, the following is stated: \begin{itemize} \item[(4)] Suppose that both \(X\), \(Y\) are topological spaces and \(f:(X,\tau)\rightarrow (Y,\mu)\) is a surjective mapping. Then the following are equivalent. \begin{itemize} \item[(i)] The topology \(\mu\) of the space \(Y\) is the finest topology that makes \(f\) preserve \(\mathcal{I}\)-convergence. \item[(ii)] \(\mu=\tau_{f,\mathcal{I}_{sn}}\), where \(\tau_{f,\mathcal{I}_{sn}}=\{U\subset Y:f^{-1}(U) \ \text{is an} \ \mathcal{I}_{sn}\text{-open subset of} \ X\}.\) \item[(iii)] The mapping \(f\) is an \(\mathcal{I}_{sn}\)-quotient mapping and \(\mu=\mu_{\mathcal{I}_{sn}}\). \end{itemize} \end{itemize} Next, given a topological space \((X,\tau)\) and \(f:X\rightarrow Y\) a surjective mapping, the author investigates how to characterize a topology \(\mu\) on the set \(Y\) such that it is the finest topology that makes \(f:(X,\tau)\rightarrow (Y,\mu)\) \(\mathcal{I}\)-continuous. For that, the concepts of \(\mathcal{I}\)-open topology, denoted by \(\tau_{\mathcal{I}}\), and an \(\mathcal{I}\)-open topological space, denoted by \(X_{\mathcal{I}}\), are defined. The following is another important result of the paper. \begin{itemize} \item[(5)] Let \(X\) be an \(\mathcal{I}\)-topological space, \(f:X\rightarrow Y\) be a surjective mapping and \((Y, \tau_{f,\mathcal{I}})\), where \(\tau_{f,\mathcal{I}}=\{U\subset Y :f^{-1}(U) \ \text{is} \ \mathcal{I}\text{-open in} \ X\}\), be an \(\mathcal{I}\)-sequential space. Then the following are equivalent: \begin{itemize} \item[(i)] The topology \(\mu\) of the space \(Y\) is the finest topology that makes \(f:(X,\tau)\rightarrow(Y,\mu)\) \(\mathcal{I}\)-continuous. \item[(ii)] \(\mu=\tau_{f,\mathcal{I}}\). \item[(iii)] The mapping \(f:(X,\tau)\rightarrow(Y,\mu)\) is an \(\mathcal{I}\)-quotient mapping and \(Y\) is an \(\mathcal{I}\)-sequential space. \end{itemize} \end{itemize} Corollaries and examples that strengthen the results of the paper complete the whole study of the paper.
scientific article; zbMATH DE number 7381185
scientific article