Maximal topologies with respect to a family of discrete subsets (Q2329356)

In this paper, some characterizations of maximal spaces with respect to the family of discrete subsets are studied. The following Theorem is proved: ``A space \((X,\tau)\) is maximal with respect to the family of its discrete subsets if and only if for every \(A\subset X\), \((\partial ^{\ast ,\tau}A)\cap A\) is discrete''. Also: ``There is a topological Hausdorff space \((X,\tau)\) that is not maximal with respect to the family of discrete subsets'' is introduced as an example in the paper.