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.

A progressive approach to multi-criteria group decision-making: N-bipolar hypersoft topology perspective

“…HS sets employ a multi-argument approximation function strategy, enhancing adaptability and reliability compared to traditional soft sets. Building on this, Musa and Asaad [ 39 ] introduced bipolar hypersoft (BHS) sets demonstrating their applicability in decision-making problems [ 40 ]. Their subsequent work delved into the topological structures associated with BHS sets [ 41 , 42 ].…”

“…N -HS sets integrate a parametrized object characterization based on ordered grades, providing a versatile representation for intricate real-world scenarios. Furthermore, they [ 49 ] introduced the notion of N -HST, extending the concept from HST [ 50 ].…”

N-Hypersoft Sets: An Innovative Extension of Hypersoft Sets and Their Applications

“…Extensions of HS sets, including plithogenic HS sets [38], complex fuzzy HS sets [39], interval-valued intuitionistic fuzzy HS sets [40], and complex Q-rung orthopair fuzzy hypersoft sets [41] have been investigated, and practical applications have been discussed. Musa and Asaad developed the idea of bipolar HS sets, as documented in their work [42], and used their applications in decision-making [43] and topological concepts [44].…”

Soft C-continuity and soft almost C-continuity between soft topological spaces

“…Soft topological concepts were established by Shabir and Naz to be equivalent to their counterparts in classical topology, which motivated and supported researchers in the field to continue in this direction. Numerous contributions have been made to the exploration of topological notions and concepts in soft environments since the development of soft topology [8] , [9] , [10] , [11] , [12] , [13] , [14] , [15] , [16] , [17] , [18] , [19] , [20] , [21] , [22] , [23] , [24] , [25] , [26] , [27] , [28] , [29] , [30] .…”