Generalized submaximal spaces
According to \textit{Á. Császár} [Acta Math. Hung. 96, No. 4, 351--357 (2002; Zbl 1006.54003)], a generalized topology on a set \(X\) is a family \(\mu\) of subsets of \(X\) such that \(\emptyset \in \mu\) and the arbitrary union of members of \(\mu\) belongs to \(\mu\). The elements of \(\mu\) are called \(\mu\)-open sets. In a straightforward manner, \(\mu\)-closed sets can be defined as well as the \(\mu\)-interior and the \(\mu\)-closure of an arbitrary subset \(S\subseteq X\). As a consequence, the definitions of semi-open sets and preopen sets can be extended to the context of \(\mu\)-open sets. The present paper investigates the situation where a generalized space \((X,\mu )\) is submaximal, i.e. every \(\mu\)-dense set is \(\mu\)-open. It turns out that several results from the topological setting are also valid for generalized spaces. In particular, a generalized space \((X,\mu )\) is submaximal if and only if every subset of \(X\) is \(\mu\)-locally closed.
generalized extremally disconnected space
scientific article; zbMATH DE number 6006172