On the Relationship between Ideal Cluster Points and Ideal Limit Points

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Let X be a first countable space which admits a non-trivial convergent sequence 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 without isolated points. 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≠ is regular closed, 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.