Ideal-quasi-Cauchy sequences

TL;DR

This paper introduces and studies the concepts of $I$-ward compactness and $I$-ward continuity, generalizing classical notions using ideal-based convergence and quasi-Cauchy sequences.

Contribution

It defines $I$-ward compactness and $I$-ward continuity, establishing their relationships with existing forms of compactness and continuity in real analysis.

Findings

$I$-ward compact sets have specific subsequence properties.

$I$-ward continuous functions preserve $I$-quasi-Cauchy sequences.

Connections between $I$-ward concepts and classical compactness and continuity.

Abstract

An ideal $I$ is a family of subsets of positive integers $\textbf{N}$ which is closed under taking finite unions and subsets of its elements. A sequence $(x_n)$ of real numbers is said to be $I$-convergent to a real number $L$, if for each \;$ \varepsilon> 0$ the set $\{n:|x_{n}-L|\geq \varepsilon\}$ belongs to $I$. We introduce $I$-ward compactness of a subset of $\textbf{R}$, the set of real numbers, and $I$-ward continuity of a real function in the senses that a subset $E$ of $\textbf{R}$ is $I$-ward compact if any sequence $(x_{n})$ of points in $E$ has an $I$-quasi-Cauchy subsequence, and a real function is $I$-ward continuous if it preserves $I$-quasi-Cauchy sequences where a sequence $(x_{n})$ is called to be $I$-quasi-Cauchy when $(\Delta x_{n})$ is $I$-convergent to 0. We obtain results related to $I$-ward continuity, $I$-ward compactness, ward continuity, ward compactness, ordinary compactness, ordinary continuity, $\delta$-ward continuity, and slowly oscillating continuity.