Rough Families, Cluster Points, and Cores

We define the notion of ideal convergence for sequences $(x_n)$ with values in topological spaces $X$ with respect to a family $\{F_\eta: \eta \in X\}$ of subsets of $X$ with $\eta \in F_\eta$. Each set $F_\eta$ quantifies the degree of accuracy of the convergence toward $\eta$. 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 $(x_n)$.