When degree of roughness is a neighborhood over locally solid Riesz spaces

TL;DR

This paper introduces generalized notions of rough weighted limit and cluster points in locally solid Riesz spaces, compares their properties with classical results, and explores their implications for summability methods.

Contribution

It extends the concept of rough limit points to locally solid Riesz spaces and analyzes their properties, including conditions under which they are closed or not.

Findings

Weighted $\ ext{I}_\tau$-cluster points may not be closed.

In locally solid Riesz spaces, the classical closedness results do not always hold.

The new summability method generalizes previous results in the literature.

Abstract

In this paper we introduce the notion of rough weighted $\mathcal{I}_\tau$-limit points set and weighted $\mathcal{I}_\tau$-cluster points set in a locally solid Riesz space which are more generalized version of rough weighted $\mathcal{I}$-limit points set and weighted $\mathcal{I}$-cluster points set in a $\theta$-metric space respectively. Successively to compare with the following important results of Fridy [Proc. Amer. Math. Soc. {118} (4) (1993), 1187-1192] and Das [Topology Appl. {159} (10-11) (2012), 2621-2626], respectively be stated as \begin{description} \item[(i)] Any number sequence $x=\{x_{n}\}_{n\in \mathbb{N}},$ the statistical cluster points set of $x$ is closed, \item[(ii)] In a topological space the $\mathcal{I}$-cluster points set is closed, \end{description} we show that in general, the weighted $\mathcal{I}_\tau$-cluster points set in a locally solid Riesz space may not be closed. The resulting summability method unfollows some previous results in the direction of research works of Aytar [Numer. Funct. Anal. Optim. {29} (3-4) (2008) 291-303], D$\ddot{\mbox{u}}$ndar [Numer. Funct. Anal. Optim. {37} (4) (2016) 480-491], Ghosal [Math. Slovaca {70} (3) (2020) 667-680] and Sava\c{s}, Et [Period. Math. Hungar. 71 (2015) 135-145].