A new perspective on ideal localized sequences within $\mathscr{A}$-metric spaces

Abstract

This study presents and examines the perspectives of $\mathfrak{I}$-localized sequences and $\mathfrak{I}$-Cauchy sequences in the setting of an $\mathscr{A}$-metric space. We examine the interplay between these notions under suitable conditions. The study further extends to $\mathfrak{I}$-boundedness, where we establish a significant relationship with $\mathfrak{I}$-localized sequences. We define and explore the constructs of $\mathfrak{I}$-limit points and $\mathfrak{I}$-cluster points of sequences, emphasizing their connections with the $\mathfrak{I}$-locator in an $\mathscr{A}$-metric space. The foundations of an $\mathfrak{I}$-barrier for a sequence is also explored, and we utilize it to draw important links between $\mathfrak{I}$-Cauchy and $\mathfrak{I}$-localized sequences. Furthermore, we establish connections based on the presence of $\mathfrak{I}$-nonthin Cauchy subsequences and thick subsets in the underlying space. Finally, we investigate the relationship between $\mathfrak{I}$-localized and $\mathfrak{I}^*$-localized sequences, offering new insights within the context of $\mathscr{A}$-metric spaces.