Let $\mathcal {P}(\mathbf{N})$ be the power set of N. We say that a function $\mu ^\ast : \mathcal {P}(\mathbf{N}) \to \mathbf{R}$ is an upper density if, for all X, Y ⊆ N and h, k ∈ N⁺, the following hold: (f1) $\mu ^\ast (\mathbf{N}) = 1$ ; (f2) $\mu ^\ast (X) \le \mu ^\ast (Y)$ if X ⊆ Y; (f3) $\mu ^\ast (X \cup Y) \le \mu ^\ast (X) + \mu ^\ast (Y)$ ; (f4) $\mu ^\ast (k\cdot X) = ({1}/{k}) \mu ^\ast (X)$ , where k · X : = {kx: x ∈ X}; and (f5) $\mu ^\ast (X + h) = \mu ^\ast (X)$ . 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 $\mu ^\ast (X)\le 1$ 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 {\mathcal{I}} on ω, we prove that a sequence in a topological space X is {\mathcal{I}} -convergent if and only if there exists a “big” {\mathcal{I}} -convergent subsequence. Then we study several properties and show two characterizations of the set of {\mathcal{I}} -cluster points as classical cluster points of a filter 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 {(\tau,\mathcal{I})} .

Project Euclid - mathematics and statistics online