Sum of Soft Topological Spaces

Abstract

In this paper, we introduce the concept of sum of soft topological spaces using pairwise disjoint soft topological spaces and study its basic properties. Then, we define additive and finitely additive properties which are considered a link between soft topological spaces and their sum. In this regard, we show that the properties of being p-soft T i , soft paracompactness, soft extremally disconnectedness, and soft continuity are additive. We provide some examples to elucidate that soft compactness and soft separability are finitely additive; however, soft hyperconnected, soft indiscrete, and door soft spaces are not finitely additive. In addition, we prove that soft interior, soft closure, soft limit, and soft boundary points are interchangeable between soft topological spaces and their sum. This helps to obtain some results related to some important generalized soft open sets. Finally, we observe under which conditions a soft topological space represents the sum of some soft topological spaces.

New Soft Structure: Infra Soft Topological Spaces

“…is concept was independently reformulated by Samanta et al [11,12], while Das and Samanta [11] applied the new version of the soft point to study the concept of soft metric spaces and Nazmul and Samanta [12] used it to discuss soft neighbourhood systems and reveal some relations of soft limit points of a soft set. Many scholars analyzed the properties of soft topologies and compared their performance with the case of classical topologies, see, for example, [13][14][15][16][17][18][19][20][21][22][23]. Generalizations of open sets were investigated in soft topologies, see [24,25].…”

Belong and Nonbelong Relations on Double-Framed Soft Sets and Their Applications

“…After Molodtsov's work, many researchers have studied several operations and relations between soft sets (see, for example, [5][6][7][8][9][10]). Soft sets were applied in various domains such as algebraic structures (see, for example, [11][12][13]), soft topological spaces (see, for example, [14][15][16]), and decision-making problems (see, for example, [17][18][19][20][21][22][23][24][25]). Also, the relationship among soft sets, rough sets, and fuzzy sets was the goal of some papers such as [17,26,27].…”

Some Modifications of Pairwise Soft Sets and Some of Their Related Concepts

“…For a STS (U, τ, E), the members τ are called soft open sets. Soft topological concepts and their applications are still a hot area of research [1,2,[5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. The concept of ω-open sets in TSs is defined in [23] as follows: let (U, ) be a TS and V ⊆ U, then V is ω-open set in (U, ) if for each x ∈ V, there is W ∈ such that x ∈ W and W − V is countable, or equivalently, V is ω-open set in (U, ) if and only if for each x ∈ V, there is W ∈ and a countable set C ⊆ U such that x ∈ W −C ⊆ V. Denote the family of all ω-open sets in the TS (U, ) by ω .…”