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The main objective of this research is to investigate and introduce a novel concept of soft semi-open sets, named soft -open sets. Some of their properties and relationship with other types of soft sets were explored and discussed. Introduction Molodtsov [1] primarily introduced the concept of soft sets as an entirely new approach to deal with imperfect information and has since been effectively implemented in many areas including functions smoothness and other related theories. Formulation of soft operators was first attempted and presented by Maji et al. [2], describing null and absolute soft sets as complements of a soft set, a soft union and a soft intersection between two soft sets. The main concepts of soft topologies were investigated by Shabir and Naz [3] for further formulations including soft closure operators, soft subspaces, and soft separation axioms, and followed by Hussain and Ahmad [4] investigations of soft boundary operators and soft interior. Chen [5] defined the properties of soft semi-open sets and soft semi-continuous maps. Ilango and Ravindran [6] introduced and defined the properties of soft pre-open, soft regular sets and soft pre-closed sets. The concept of soft semi-open sets and soft closed sets was used by Khalaf and Musa [7] to introduce define new types of semi-open sets denoted -open sets. Fayad and Mahdi [8] described soft -open sets utilizing the notions of soft -open and soft closed sets. Based on the properties of soft pre-open and soft closed sets, Ahmed and Hamko [9] defined the notion of soft -open sets. Mohammed and et. al. [10,11] defined and proposed new types of soft open sets (resp., soft regular open) sets and denoted them soft -open (resp., soft -open) sets. The notion of soft -open sets was finally presented by Hameed and Hussein [12], using soft b-open and soft closed sets. However, this work introduces a new type of soft sets called soft -open using semi-open set and soft pre-closed sets, which is strictly placed between the classes of soft -open sets and soft semi-open sets. Some of its basic properties and relationships with some other types of soft sets are given. |
