Connectedness of Soft r-Topological Spaces

Abstract

Topological properties are essential for developing digital line spaces and computer graphics other than the field of topology. Soft topological properties must play an equal role as classical topological properties or even better. Soft connectedness is one of the most fundamental soft topological aspects. It studies the nearness of two objects from a topological point of view. By considering the significance of this concept, we make this contribution to study the connectedness of some strong soft topologies. We begin by introducing a new class of soft open sets, named soft [Formula: see text]-open sets, followed by establishing its main properties. We show that the collection of all soft [Formula: see text]-open sets constitutes a soft topology, which is coarser than the original one. Then, we define the concept of soft [Formula: see text]-separated sets, which helps us to give the [Formula: see text]-connectedness of a soft set. We show that soft connectedness implies soft [Formula: see text]-connectedness, which implies soft [Formula: see text]-connectedness of a soft set. Counterexamples are provided to show that the implications are not reversible. However, they are identical on a soft open set. Further properties and characterizations of soft [Formula: see text]-connected sets are proposed.