Let P(N) be the power set of N. We say that a function µ ⋆ : P(N) → R is an upper density if, for all X, Y ⊆ N and h, k ∈ N + , the following hold: (f1) µ ⋆ (N) = 1; (f2)We show that the upper asymptotic, upper logarithmic, upper Banach, upper Buck, upper Pólya, and upper analytic densities, together with all upper α-densities (with α a real parameter ≥ −1), are upper densities in the sense of our definition. Moreover, we establish the mutual independence of axioms (f1)-(f5), and we investigate various properties of upper densities (and related functions) under the assumption that (f2) is replaced by the weaker condition that µ ⋆ (X) ≤ 1 for every X ⊆ N.Overall, this allows us to extend and generalize results so far independently derived for some of the classical upper densities mentioned above, thus introducing a certain amount of unification into the theory.

Given an ideal I on ω, we prove that a sequence in a topological space X is I-convergent if and only if there exists a "big" I-convergent subsequence. Then, we study several properties and show two characterizations of the set of I-cluster points as classical cluster points of a filters on X and as the smallest closed set containing "almost all" the sequence. As a consequence, we obtain that the underlying topology τ coincides with the topology generated by the pair (τ, I).2010 Mathematics Subject Classification. Primary: 40A35, 54A20. Secondary: 11B05.

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.2010 Mathematics Subject Classification. Primary: 40A35. Secondary: 54A20, 40A05, 11B05.

We define a class of so-called thinnable ideals I on the positive integers which includes several well-known examples, e.g., the collection of sets with zero asymptotic density, sets with zero logarithmic density, and several summable ideals. Given a sequence (xn) taking values in a separable metric space and a thinnable ideal I, it is shown that the set of I-cluster points of (xn) is equal to the set of I-cluster points of almost all its subsequences, in the sense of Lebesgue measure.Lastly, we obtain a characterization of ideal convergence, which improves the main result in [Trans.

Let S be the set of subsequences (xn k ) of a given real sequence (xn) which preserve the set of statistical cluster points. It has been recently shown that S is a set of full (Lebesgue) measure. Here, on the other hand, we prove that S is meager if and only if there exists an ordinary limit point of (xn) which is not also a statistical cluster point of (xn). This provides a non-analogue between measure and category.2010 Mathematics Subject Classification. Primary: 40A35. Secondary: 11B05, 54A20.