Separation Axioms on Bipolar Hypersoft Topological Spaces

Abstract

According to its definition, a topological space could be a highly unexpected object. There are spaces (indiscreet space) which have only two open sets: the empty set and the entire space. In a discrete space, on the other hand, each set is open. These two artificial extremes are very rarely seen in actual practice. Most spaces in geometry and analysis fall somewhere between these two types of spaces. Accordingly, the separation axioms allow us to say with confidence whether a topological space contains a sufficient number of open sets to meet our needs. To this end, we use bipolar hypersoft (BHS) sets (one of the efficient tools to deal with ambiguity and vagueness) to define a new kind of separation axioms called BHS Ti-space (i = 0, 1, 2, 3, 4). We show that ee BHS Ti-space (i = 1,2) implies BHS Ti−1-space; however, the converse is false, as shown by an example. e For i = 0, 1, 2, 3, 4, we prove that BHS Ti -space is hypersoft (HS) Ti -space and we present a condition so that eee HS Ti-space is BHS Ti-space. Moreover, we study that a BHS subspace of a BHS Ti-space is a BHS Ti-space for i = 0,1,2,3.