The \(H\)-sets in the unit circle are properly \(G_{\delta\sigma}\) (Q1322628)

Let \(K(I)\) denote the collection of compact subsets of the unit interval \(I\). \(E\in K(I)\) is an \(H\)-set iff there exist an interval \(J\subset I\) and infinitely many \(n\in N\) for which \(J\cap (n\cdot E)= \emptyset\), where \(n\cdot E= \{nx- [nx]\): \(x\in E\}\). In the present paper it is shown that the family of all \(H\)-sets is a properly \(G_{\delta\sigma}\) set (i.e.\(G_{\delta \sigma}\) but not \(F_{\sigma \delta}\)) in the space \(K(I)\) endowed with the Vietoris topology.