On generalizations of \(H\)-closed spaces (Q1917737)

The authors use the concept of a pre \(T_2\) space to generalize the concept of H-closed spaces. A set \(A\subset X\) is said to be preopen if \(A\subset \text{int cl }A\). A space \(X\) is said to be pre \(T_2\) if for every pair of distinct points \(x\) and \(y\) of \(X\) there exist disjoint preopen sets \(U\) and \(V\) containing \(x\) and \(y\) respectively. A subset \(A\) of a space \(X\) is said to be pre \(T_2\)-closed relative to \(X\) if and only if every preopen cover of \(A\) has a finite subfamily whose closures are a cover of \(A\). \(A\) is said to pre \(T_2\) closed if \(A\) is pre \(T_2\) closed in its relative topology. In the main section of the paper the authors prove the existence of a projective maximum and projective minimum in the class of one-point extensions for a locally pre \(T_2\)-closed extremally disconnected Hausdorff space.