The purpose of this paper is to study the effect of mappings, some decompositions of pairwise continuity and some generalized pairwise open mappings on pairwise nearly Lindelöf spaces. The main result indicates that a pairwise $\delta$-continuous image of a pairwise nearly Lindelöf space is pairwise nearly Lindelöf.

In this paper we introduce the concept of $\mathrm{rsg}$-closed sets and investigate some of its properties in topological spaces. We also define an $\mathrm{rsg}$-regular space and give some of its fundamental properties.

Let $\tau$ be any preradical and $M$ any module. In [2], Al-Takhman, Lomp and Wisbauer defined $\tau$-supplemented module. In this paper we introduce the (completely) $\tau$-$\oplus$-supplemented modules. It is shown that (1) Any finite direct sum of $\tau$-$\oplus$-supplemented modules is $\tau$-$\oplus$-supplemented. (2) If $M$ is $\tau$-$\oplus$-supplemented module and $\left(D_3\right)$ then $M$ is completely $\tau$-$\oplus$-supplemented.