The homeomorphic transformation of c-sets into superdense sets (Q1262964)

Let \(I=[0,1]\). If every \(x\in E\subseteq I\) is a bilateral condensation point of E then E is called c-set. Let p: \(I\to I\) be a nondecreasing function with \(\lim_{h\to 0^+}p(h)=0\). Then E is said p-dense at x if \(\lim_{h\to 0^+}m((x-h,x+h)\cap (I-E))/p(h)=0\). E is said p-set if E is p-dense at each of its points. It is proven the following Theorem: Let E be an \(F_{\sigma}\) c-set in I, and suppose \(p:\quad I\to I\) is nondecreasing and \(\lim_{h\to 0^+}p(h)=0\). Then there is a homeomorphism \(H:\quad I\to I\) that transforms E into an \(F_{\sigma}\) p-set.