Rough families, cluster points, and cores

We define the notion of ideal convergence for sequences (xn) with values in topological spaces X with respect to a family {Fη : η ∈ X} of subsets of X with η ∈ Fη. Each set Fη quantifies the degree of accuracy of the convergence toward η. After proving that this is really a new notion, we provide some properties of the set of limit points and characterize the latter through the ideal cluster points and the ideal core of (xn).