On the relationship between ideal cluster points and ideal limit points

Abstract

Let X be a first countable space which has a non-isolated point and let I be an analytic P-ideal. First, it is shown that the sets of I-limit points of all sequences in X are closed if and only if I is also an Fσ-ideal.Moreover, let (xn) be a sequence taking values in a Polish space. It is known that the set A of its statistical limit points is an Fσ-set, the set B of its statistical cluster points is closed, and that the set C of its ordinary limit points is closed, with A⊆B⊆C. It is proved the sets A and B own some additional relationship: indeed, the set S of isolated points of B is contained also in A.Conversely, if A is an Fσ-set, B is a closed set with a subset S of isolated points such that B∖S≠∅, and C is a closed set with S⊆A⊆B⊆C, then there exists a sequence (xn) for which: A is the set of its statistical limit points, B is the set of its statistical cluster points, and C is the set of its ordinary limit points.Lastly, we discuss topological nature of the set of I-limit points when I is neither Fσ- nor analytic P-ideal.