Although the concept of connectedness may seem simple, it holds profound implications for topology and its applications. The concept of connectedness serves as a fundamental component in the Intermediate Value Theorem. Connectedness is significant in various applications, including geographic information systems, population modeling and robotics motion planning. Furthermore, connectedness plays a crucial role in distinguishing between different topological spaces. In this paper, we define soft weakly connected sets as a new class of soft sets that strictly contains the class of soft connected sets. We characterize this new class of sets by several methods. We explore various results related to soft subsets, supersets, unions, intersections and subspaces within the context of soft weakly connected sets. Additionally, we provide characterizations for soft weakly connected sets classified as soft pre-open, semi-open or $\alpha$-open sets. Furthermore, we introduce the concept of a soft weakly connected component as follows: Given a soft point $a_x$ in a soft topological space $\left(X,\mathrm{\Delta },A\right)$, we define the soft weakly component of $\left(X,\mathrm{\Delta },A\right)$ determined by $a_x$ as the largest soft weakly connected set, with respect to the soft inclusion ( $\tilde{\subseteq }$) relation, that contains $a_x$. We demonstrate that the family of soft weakly components within a soft topological space comprises soft closed sets, forming a soft partition of the space. Lastly, we establish that soft weak connectedness is preserved under soft $\alpha$-continuity.

In the present article, a new category of mathematical structure is described based on the topological structure "primal" and the notion of "generalized". Such a structure is discussed in detail in terms of topological properties and some basic theories. Also, we introduced some operators using the concepts "primal" and "generalized primal neighbourhood", which have a lot of nice properties.