Compact and locally compact spaces via the class of soft δ-open sets

One popular approach to studying topological concepts is to employ a subclass of topology, such as clopen, regular open, and $\delta$-open sets. Motivated by the advantages of soft topology over classical topologies, we investigate some of these classes in a soft setting. We begin this manuscript by exploring further properties of soft regular open and soft $\delta$-open sets in the context of soft subspaces and soft mappings, as well as describing their behavior in relation to classical topologies. We demonstrate that the condition of an extended soft topology guarantees the symmetry between $\delta$-open sets and soft $\delta$-open sets the realms of soft topology and its crisp topologies. Then, we apply the concept of soft $\delta$-open sets to establish two new classes of soft compactness, namely soft $\delta$-compactness, and soft local $\delta$-compactness. We research the basic properties of these classes, including characterizations and preservation theorems under soft $\delta$-continuous mappings. We reveal the relationship between soft compactness, soft $\delta$-compactness and soft local $\delta$-compactness, and also prove the equivalence between these concepts when the soft topology is soft regular. Finally, the symmetry between our new classes and their counterparts in some classical topologies is studied amply; especially, when the soft topology is extended or stable. The implementations of the current results and relationships are elucidated by some supporting examples.