On strongly NBD-finite families (Q1823498)

A family \(\{A_ i:\) \(i\in I\}\) of subsets of a topological space X is said to be strongly neighbourhood finite if for every \(x\in X\) there exists an open set V containing x satisfying one of the following conditions: (a) \(V\cap A_ i=\emptyset\) for every \(i\in I\); (b) There is a nonempty finite subset \({\mathcal J}\) of I such that (i) \(V\cap A_ i\neq \emptyset\) only for every \(i\in {\mathcal J}\), (ii) \(V\cap A_ i\subset A_ j\) for all \(i,j\in {\mathcal J}\). The author studies several properties of strongly neighbourhood finite families.