Roughness of nets
Abstract
In this paper, we introduce the concepts of ~`\emph{rough weighted ideal-τ-convergence}' and ~`\emph{rough weighted ideal-τ-cluster point}' of a net in locally solid Riesz spaces. We also compare these concepts with key findings from the works of Lahiri & Das (2007/2008), Das (2012) and Aytar (2008), as follows: [(i):] The set of \emph{ideal-cluster points} of a sequence in a topological space is closed, [(ii):] The set of \emph{ideal-cluster points} of a net in a topological space is closed, [(iii):] The set of \emph{rough ideal-cluster points} of a sequence in a norm linear space is closed. We demonstrate that, generally, the set of ~`\emph{rough weighted ideal-τ-cluster points}' of a net in a locally solid Riesz space may not satisfy the aforementioned results, which implies that it may not be closed. This suggests that this set exhibits new characteristics, which we will discuss here. Ultimately, our characterization demonstrates that these new concepts can help identify an ideal that is maximal and effectively represent any closed τ-neighborhood of θ within these spaces.
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BibTeX
@misc{ghosal2026roughness,
title = {{Roughness of nets}},
author = {Sanjoy Ghosal and Mandobi Banerjee},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06217},
eprint = {2611.06217},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06217},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Roughness of nets
AU - Sanjoy Ghosal
AU - Mandobi Banerjee
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06217
UR - https://arxiv.org/abs/2611.06217
AB - In this paper, we introduce the concepts of ~`\emph{rough weighted ideal-τ-convergence}' and ~`\emph{rough weighted ideal-τ-cluster point}' of a net in locally solid Riesz spaces. We also compare these concepts with key findings from the works of Lahiri & Das (2007/2008), Das (2012) and Aytar (2008), as follows: [(i):] The set of \emph{ideal-cluster points} of a sequence in a topological space is closed, [(ii):] The set of \emph{ideal-cluster points} of a net in a topological space is closed, [(iii):] The set of \emph{rough ideal-cluster points} of a sequence in a norm linear space is closed. We demonstrate that, generally, the set of ~`\emph{rough weighted ideal-τ-cluster points}' of a net in a locally solid Riesz space may not satisfy the aforementioned results, which implies that it may not be closed. This suggests that this set exhibits new characteristics, which we will discuss here. Ultimately, our characterization demonstrates that these new concepts can help identify an ideal that is maximal and effectively represent any closed τ-neighborhood of θ within these spaces.
ER -