On a problem of P. C. Hammer (Q1803357)

The authors construct a generalized closure function \(g: {\mathcal P}(\mathbb{R})\to{\mathcal P}(\mathbb{R})\) such that (i) \(g(X)\supseteq X\), (ii) \(X\subseteq Y\) implies \(g(X)\subseteq g(Y)\), (iii) \(g^ 2= g^ 3\) and (iv) \(h^ n\neq h^ m\) for \(n\neq m\in\omega\), where \(h(X)= \mathbb{R}\backslash g(X)\).