DOI: 10.15672/hujms.1837190 ISSN: 2651-477X

When rough I-convergent sequences fail to be rough I-Cauchy

Sourav Mandal, Sanjoy Ghosal
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This study builds on the research by Collins et al. [Topology Appl. 154 (2007) 2312-2322], Aytar [Numer. Funct. Anal. Optim. {29} (3-4) (2008) 291-303] and Kostyrko et al. [Real Anal. Exchange {26} (2) (2000/2001) 669-686] to introduce the concepts of forward (resp. backward) rough I-convergence, as well as forward (resp. backward) rough I-Cauchy sequences, in asymmetric metric spaces. In this context, we would like to emphasize the following points:[(i):] A forward (resp. backward) rough I-convergent sequence may not necessarily be a forward (resp. backward) rough I-Cauchy sequence.[(ii):] We address an open problem proposed by Das et al. [Page-561, Math. Slovaca, 63 (3), 2013, 545-562].Simultaneously, we introduce forward (resp. backward) rough I-limit points, as well as forward (resp. backward) rough I-cluster points, within the context of asymmetric metric spaces to characterize an ideal and the metrizability of asymmetric metric spaces. Furthermore, we demonstrate the non-existence of any sequence $x=\{x_n\}_{n\in\mathbb{N}}$ for which a forward (resp. backward) closed set $F$ does not coincide with the forward (resp. backward) rough I-cluster point set of that sequence.

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