Maximal Soft Compact and Maximal Soft Connected Topologies

Samer Al Ghour,Zanyar A Ameen,Anandakumar Haldorai

doi:10.1155/2022/9860015

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.

Highlights

General topology is the branch of topology that deals with the fundamental set-theoretic notions and constructions used in topology
Soft topology, which combines soft set theory and topology, is another field of topology. It is concerned with a structure on the set of all soft sets and is motivated by the standard axioms of classical topological space. e work of Shabir and Nazs [7], in particular, was crucial in establishing the field of soft topology
A soft topological space (Z, I, E) is called soft connected if it cannot be written as a union of two disjoint soft open sets

Summary

Preliminaries

Definition 1 (see [19, 20]). Let 􏼈(Yi, E): i ∈ I􏼉 be a family of soft sets over Z, where I is any index set. E image of a soft set (A, E)⊆􏽥 (Z, E) under f: (Z, E) ⟶ (Y, E′) is a soft subset f(A, E) (f(A), q(E)) of (Y, E′) which is given by. E inverse image of a soft set (B, E′)⊆􏽥 (Y, E′) under f is a soft subset f−1(B, E′) (f−1(B), q−1(E′)) such that. Let (Z, I, E) and (Y, T, E′) be soft topological spaces. A soft function f: (Z, I, E) ⟶ (Y, T, E′) is said to be (i) Soft continuous if the inverse image of each soft open subset of (Y, T, E′) is a soft open subset of (Z, I, E). (iii) Soft homeomorphism if it is a soft open and soft continuous bijection from (Z, I, E) onto (Y, T, E′)

Maximal Soft Topologies

Maximal Soft Connected Topologies