Rough Families, Cluster Points, and Cores

We define the notion of ideal convergence for sequences

with values in topological spaces

X X

with respect to a family

\{F_\eta: \eta \in X\} { F η : η ∈ X }

of subsets of

X X

with

\eta \in F_\eta η ∈ F η

. Each set

F_\eta F η

quantifies the degree of accuracy of the convergence toward

\eta 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

(x_n) ( x n )

.