On rough ideal convergence

TL;DR

This paper explores the concept of rough ideal convergence in topological spaces, introducing new notions of cluster and limit points, and demonstrating their distinctness from classical ideal convergence through structural analysis and examples.

Contribution

It extends the theory of ideal convergence by defining rough convergence with respect to a family of subsets, analyzing its properties, and illustrating the differences from classical convergence.

Findings

Rough ideal convergence has unique structural properties.

The paper establishes inclusion and invariance properties of rough convergence.

Examples show rough convergence differs significantly from classical 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.