Weak structures
Let \(X\) be a nonempty set and let \(P\) be its power set. A collection \(w \subset P\) is a weak structure if \(\emptyset \in w\). Let \(w\) be a weak structure on \(X\). We define \(i_w A\) to be the union of all \(w\)-open subsets of \(A\) and \(c_w A\) to be the intersection of all \(w\)-closed subsets containing \(A\). Proposition 1. Let \(w\) be a weak structure on \(X\) and let \(A \in w\). Then (i) \(A=i_w\, A\), (ii) \(A\) is \(w\)-closed \(\rightarrow A = c_w \,A\). Proposition 2. Define \(A\in \alpha (w)\) if \(A \subset \,i_w\, c_w \,i_w \,A\) \(\qquad A\in\sigma (w)\) if \(A \subset \,c_w \,i_w \,A\) \(\qquad A\in \pi (w)\) if \(A \subset\, i_w \,A\) \(\qquad A\in \rho (w)\) if \(A \subset\,\varphi_w\, A\) \(\qquad A\in \beta (w)\) if \(A \subset\, c_w \,i_w \,c_w \,A\), where \(\varphi_w \,A= c_w \,i_w \,A\cap i_w\, c_w\, A\). Then (i) \(w \subset\alpha (w)\subset\sigma(w)\subset\rho (w)\subset\beta (w)\) and (ii) \(\alpha (w)\subset\pi (w)\subset\rho (w)\) Thus, this paper discusses the properties of \(i_w\) and \(c_w\).
scientific article; zbMATH DE number 5990602