Scattered sets and gauges (Q1374550)

A set \(A\) of real numbers is \textit{right scattered} if any nonempty subset of \(A\) has a right isolated point. The main result: Let \(\delta\) be a function with positive values, defined on the set \(R\) of reals, except possibly for a countable set. Then, except for a right scattered set, every point \(x\) is the limit from the right of some sequence \(\{ x_i\}\) for which \(\delta (x_i)\) is bounded above zero. This proposition provides simple proofs of certain known results, e.g., that any countable \(G_\delta\)-set of real numbers is scattered, or some properties of symmetric derivatives. The authors exhibit six examples.