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.

Abstract. The structures (G, r ), where r is a system of ideals defined on a directed group G, play an important role in solving arithmetical problems. In this paper, we investigate how some properties of these systems are transferred in their cartesian products and their substructures. The results we obtain find an application in the study of categorical properties of these structures.

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We find non-regular Hausdorff rU-spaces (connected or not). We also prove that a countable Hausdorff connected space cannot be an rU-space.