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On rough ideal convergence

Paolo Leonetti

TL;DR

The paper extends ideal convergence by incorporating a roughness structure via a rough family $\mathscr{F}$ and studies two main notions: rough $(\mathcal{I},\mathscr{F})$-cluster points and rough $(\mathcal{I},\mathscr{F})$-limit points. It establishes inclusion relations among the various notions, proves that cluster sets are closed under certain hyperspace topologies, and analyzes when $\mathcal{I}$-convergence and $\mathcal{I}^\star$-convergence coincide, notably for $P$-ideals under the UC-property. The authors also characterize when limit points for different notions align (via $P^+$-ideals) and provide examples showing genuine differences from classical ideal convergence. Additionally, they study the continuity properties of the maps from sequences to their rough limit/cluster sets and discuss implications in the induced hyperspace topology.

Abstract

We continue the study of ideal convergence for sequences $(x_n)$ with values in a topological space $X$ with respect to a family $\{F_η:η\in X\}$ of subsets of $X$ with $η\in F_η$, where each $F_η$ measures the allowed ``roughness'' of convergence toward $η$. 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.

On rough ideal convergence

TL;DR

The paper extends ideal convergence by incorporating a roughness structure via a rough family and studies two main notions: rough -cluster points and rough -limit points. It establishes inclusion relations among the various notions, proves that cluster sets are closed under certain hyperspace topologies, and analyzes when -convergence and -convergence coincide, notably for -ideals under the UC-property. The authors also characterize when limit points for different notions align (via -ideals) and provide examples showing genuine differences from classical ideal convergence. Additionally, they study the continuity properties of the maps from sequences to their rough limit/cluster sets and discuss implications in the induced hyperspace topology.

Abstract

We continue the study of ideal convergence for sequences with values in a topological space with respect to a family of subsets of with , where each measures the allowed ``roughness'' of convergence toward . 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.
Paper Structure (6 sections, 13 theorems, 47 equations)

This paper contains 6 sections, 13 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Rough ideal cluster points
  4. Rough ideal limit points
  5. Relationships
  6. Proofs

Key Result

Proposition 1.3

Let $\mathcal{I}$ be an ideal on $\omega$. Let also $X$ be a metric space with the UC-property and $\mathscr{F}$ be a rough family of closed sets such that there exist $\eta\in X$ and a sequence $\bm{y}\in X^\omega$ for which $\lim_k y_k=\eta$ and $y_k \notin F_\eta$ for all $k \in \omega$. Then $\m

Theorems & Definitions (46)

  • Definition 1.1
  • Definition 1.2
  • Proposition 1.3
  • Definition 2.1
  • Theorem 2.2
  • Remark 2.3
  • Example 2.4
  • Corollary 2.5
  • Corollary 2.6
  • Theorem 2.7
  • ...and 36 more