On rough ideal convergence

Abstract

We continue the study of ideal convergence for sequences $(x_n)$ with values in a topological space $X$ with respect to a family $\{F_\eta:\eta\in X\}$ of subsets of $X$ with $\eta\in F_\eta$, where each $F_\eta$ measures the allowed ``roughness'' of convergence toward $\eta$. More precisely, after introducing the corresponding notions of cluster and limit points, we prove several inclusion and invariance properties, discuss their structural properties, and give examples showing that the rough notions are genuinely different from the classical ideal ones.