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

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].