Maximal Soft Compact and Maximal Soft Connected Topologies

Abstract

In various articles, it is said that the class of all soft topologies on a common universe forms a complete lattice, but in this paper, we prove that it is a complete lattice. Some soft topologies are maximal, and some are minimal with respect to specific soft topological properties. We study the properties of soft compact and soft connected topologies that are maximal. Particularly, we prove that a maximal soft compact topology has identical families of soft compact and soft closed sets. We further show that a maximal soft compact topology is soft T 1 , while a maximal soft connected topology is soft T 0 . Lastly, we establish that each soft connected relative topology to a maximal soft connected topology is maximal.

On soft submaximal spaces

“…(1) (𝑍,  , Λ) is soft submaximal; (2) The soft boundary of any soft set over 𝑍 is soft discrete; (3) The soft boundary of any soft set over 𝑍 is soft closed; (4) Each soft co-dense set over 𝑍 is soft closed.…”

“…The class of soft submaximal spaces was defined by Ilango and Ravindran [21] , but not much study was done to further investigate soft submaximal spaces. The topic is crucial for considering the maximal element in the set of all soft topologies (see [2] ) or soft topological groups on a common universe, and for extending many other different results done in [10] . Hence, we develop the properties of soft submaximal spaces and obtain several of their characterizations.…”

“…The class of soft submaximal spaces was defined by Ilango and Ravindran [21], but not much study was done to further investigate soft submaximal spaces. The topic is crucial for considering the maximal element in the set of all soft topologies (see [2]) or soft topological groups on a Definition 2.1. [18,30] A soft topology on 𝑍 is a collection  of  Λ (𝑍) if  meets the following conditions:…”

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] .…”

Bipolar Soft Generalized Topological Structures and Their Application in Decision Making

“…Likewise, for the demonstration of the notion of soft topological spaces, the soft sets were also, C ¸aǧman [15]. This was followed by a excess of researches that tackled the soft topological spaces (see [2], [3], [4], [5], [6], [7], [8], [10], [11], [12], [18], [19], [22], [23], [25], [26], [33], [38], [40]).…”