Separation axioms and fixed points using total belong and total non-belong relations with respect to soft β-open sets

Abstract

One of the divergences between soft set and crisp set is the diversity of belong and non-belong relations between soft sets and ordinary points. This widely opens the door to establish several classes of separation axiom on soft setting. In this article, we exploit the relations of total belong and total non-belong to introduce new soft separation axioms with reference to ordinary points, namely tt-soft βTi (i = 0, 1, 2, 3, 4) and tt-soft β -regular spaces. The advantages behind using these relations are, first, generalization of existing comparable properties on general topology and second eliminating the stability shape of soft open and closed subsets of soft β -regular spaces. With the aid of some examples, we show the relationships between them as well as with soft βTi (i = 0, 1, 2, 3, 4) and soft β -regular spaces. Also, we explain the role of soft hyperconnected and extended soft topological spaces in obtaining some interesting results. We characterize a tt-soft β -regular space and demonstrate that it guarantees the equivalence of tt-soft βTi (i = 0, 1, 2). Further, we investigate these soft separation axioms in terms of product soft spaces and sum of soft topological spaces. Finally, we introduce a concept of β -fixed soft point theorem and study its main properties.