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 $(x_n)$.