Finite soft-open sets: characterizations, operators and continuity

In this paper, we present a novel family of soft sets named "finite soft-open sets". The purpose of investigating this kind of soft sets is to offer a new tool to structure topological concepts that are stronger than their existing counterparts produced by soft-open sets and their well-known extensions, as well as to provide an environment that preserves some topological characteristics that have been lost in the structures generated by celebrated extensions of soft-open sets, such as the distributive property of a soft union and intersection for soft closure and interior operators, respectively. We delve into a study of the properties of this family and explore its connections with other known generalizations of soft-open sets. We demonstrate that this family strictly lies between the families of soft-clopen and soft-open sets and derive under which conditions they are equivalent. One of the unique features of this family that we introduce is that it constitutes an infra soft topology and fails to be a supra soft topology. Then, we make use of this family to exhibit some operators in soft settings, i.e., soft $fo$-interior, $fo$-closure, $fo$-boundary, and $fo$-derived. In addition, we formulate three types of soft continuity and look at their main properties and how they behave under decomposition theorems. Transition of these types between realms of soft topologies and classical topologies is examined with the help of counterexamples. On this point, we bring to light the role of extended soft topologies to validate the properties of soft topologies by exploring them for classical topologies and vice-versa.