Soft $\omega$-almost-regularity, soft $\omega$ -semi-regularity, and soft $\omega$- $T_{2\frac{1}{2}}$ as three novel soft separation axioms are introduced. It is demonstrated that soft $\omega$ -almost-regularity is strictly between "soft regularity" and "soft almost-regularity"; soft $\omega$- $T_{2\frac{1}{2}}$ is strictly between "soft $T_{2\frac{1}{2}}$" and "soft $T_2$", and soft $\omega$ -semi-regularity is a weaker form of both "soft semi-regularity" and "soft $\omega$-regularity". Several sufficient conditions for the equivalence between these new three notions and some of their relevant ones are given. Many characterizations of soft $\omega$-almost-regularity are obtained, and a decomposition theorem of soft regularity by means of "soft $\omega$ -semi-regularity" and "soft $\omega$-almost-regularity" is obtained. Furthermore, it is shown that soft $\omega$-almost-regularity is heritable for specific kinds of soft subspaces. It is also proved that the soft product of two soft $\omega$-almost regular soft topological spaces is soft $\omega$-almost regular. In addition, the connections between our three new conceptions and their topological counterpart topological spaces are discussed.

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.

This paper introduces a novel concept of supra-soft topology generated from a specific family of supra-topologies. We define supra-soft $\omega$-open sets as a new class of soft sets that create a finer topology than the original. We explore the properties of these supra-soft $\omega$-open sets and assess the validity of related results from ordinary supra-topological spaces within the framework of supra-soft topological spaces. Additionally, we present two new separation axioms: supra-soft $\omega$-local indiscreteness and supra-soft $\omega$ -regularity, demonstrating that both are stronger than traditional supra-soft $\omega$-regularity. We also provide subspace and product theorems for supra-soft $\omega$-regularity and examine the correspondence between our new supra-soft concepts and their classical counterparts in supra-topology.

Beyond the realm of soft topology, soft continuity can aid in the creation of digital images and computational topological applications. This paper investigates soft almost weakly continuous, a novel family of generalized soft continuous functions. The soft pre-continuous and soft weakly continuous function classes are included in this class. We obtain many characterizations of soft almost weakly continuous functions. Furthermore, we investigate the link between soft almost weakly continuous functions and their general topology counterparts. We present adequate conditions for a soft almost weakly continuous function to become soft weakly continuous (soft pre-continuous). We also present various results of soft composition, restriction, preservation, product, and soft graph theorems in terms of soft almost weakly continuous functions.

We continued the study of "soft strong $\theta$-continuity" and defined and investigated "soft almost strong $\theta$-continuity" which is a generalization of soft strong $\theta$-continuity. We gave characterizations and examined soft composition concerning these two concepts. Furthermore, we derived several soft mapping theorems. We provided several links between these two ideas and their related concepts through examples. Lastly, we looked at the symmetry between them and their topological counterparts.

This paper introduces two new weak forms of soft regularity in soft topology: soft $p$-regular and soft almost- $p$-regular. These forms extend their analogous notions in classical topological spaces by incorporating soft sets, providing a flexible framework for dealing with uncertainty or imprecision. We obtain several characterizations of each of them and investigate the correspondences between them and their analogous concepts in classical topological spaces. Also, we prove that soft $p$ -regularity lies strictly between soft regularity and soft almost- $p$ -regularity. Moreover, we show that soft $p$-regularity (resp. soft almost $p$-regularity) is preserved under soft $\alpha$-open subspaces (resp. soft regular-open subspaces). In addition, we prove that soft $p$-regularity and soft almost $p$-regularity are soft productive. Finally, we obtain some preservation theorems regarding soft $p$-regularity and soft almost- $p$-regularity.

Firstly, for the topological spaces that contain a minimal open set, we obtain various inclusions between minimal open sets and homogeneity components. For a given soft topological space $(X,\tau ,A)$, we define soft homogeneous components. We show that soft homogeneous components of $(X,\tau ,A)$ form a soft partition of the absolute soft set. Also, we show that $(X,\tau ,A)$ is soft homogeneous if and only if it has only one soft homogeneous component. Moreover, we study the relationships between the soft homogeneous components of $(X,\tau ,A)$ and the homogeneous component. For the soft topological spaces that contain a minimal soft open set, we obtain various inclusions between minimal soft open sets and soft homogeneity components. In addition, we show that soft homeomorphisms stabilize soft homogeneous components. Additionally, we introduce two soft product theorems concerning soft homogeneity and soft minimality, respectively.