Cardinality of accumulation points of infinite sets

Abstract

One of the fundamental theorems in real analysis is the Bolzano-Weierstrass property according to which every bounded infinite set of real numbers has an accumulation point. Since this theorem essentially asserts the completeness of the real numbers, the notion of accumulation point becomes substantial. This work provides an efficient number of examples which cover every possible case in the study of accumulation points, classifying the different sizes of the derived set A and of the sets A ∩ A , A \ A, for an infinite set A.