Remarks on generalized hyperconnectedness
Results include the following: Theorem 1. Let \((X, \mu)\) be a strong generalized topological space. The following conditions are equivalent: (a) \(X\) is hyperconnected. (b) \(c_{\beta} (U) = X\) for every non empty semi open set \(U\), where \(c_{\beta}\) denotes the \(\beta\)-closure. (c) \(c_\pi (U) = X\) for every non empty semi open set \(U\) where \(\pi\) =\{all preopen sets\}. Theorem 2. Let \((X, \mu)\) be a generalized topological space. If \(X\) is \(\beta\)-connected, then \(X\) is \(\pi\)-irreducible. Theorem 3. Let \((X, \mu)\) be an extremally disconnected generalized topological space. Then \(X\) is strong. Theorem 4. Let \((X, \mu)\) be a strong space. If \(X\) is hyperconnected, then \(X\) is extremally disconnected. There is also a section dealing with embedding into hyperconnected spaces.
extremally disconnected and strong spaces
scientific article; zbMATH DE number 6083189