A novel class of bipolar soft separation axioms concerning crisp points

Abstract

Abstract The main objective of this study is to define a new class of bipolar soft (BS) separation axioms known as BS T ˜ ˜ i {\widetilde{\widetilde{T}}}_{i} -space ( i = 0 , 1 , 2 , 3 , 4 ) \left(i=0,1,2,3,4) . This type is defined in terms of ordinary points. We prove that BS T ˜ ˜ i {\widetilde{\widetilde{T}}}_{i} -space implies BS T ˜ ˜ i − 1 {\widetilde{\widetilde{T}}}_{i-1} -space for i = 1 , 2 i=1,2 ; however, the opposite is incorrect, as demonstrated by an example. For i = 0 , 1 , 2 , 3 , 4 i=0,1,2,3,4 , we investigate that every BS T ˜ ˜ i {\widetilde{\widetilde{T}}}_{i} -space is soft T ˜ i {\widetilde{T}}_{i} -space; and we set up a condition in which the reverse is true. Moreover, we point out that a BS subspace of a BS T ˜ ˜ i {\widetilde{\widetilde{T}}}_{i} -space is a BS T ˜ ˜ i {\widetilde{\widetilde{T}}}_{i} -space for i = 0 , 1 , 2 , 3 i=0,1,2,3 .