\(k\)-primal spaces (Q2077288)

In the context of \(k\)-primal spaces, the authors prove that a topology with finite specialization qoset \((X,\leq)\) is \(k\)-primal for some positive integer \(k\), if and only if for any cyclic point \(a\) and any \(x\) and \(y\) in \(X\) such that \(a\leq y\) and \(x\leq y\) it happens that \(a\leq x\). Also, they prove that the topologies on a finite set which are \(k\)-primal for some positive integer \(k\) form a complemented lattice.