Soft ωp-Open Sets and Soft ωp-Continuity in Soft Topological Spaces

Samer Al Ghour

doi:10.3390/math9202632

Abstract

We define soft ωp-openness as a strong form of soft pre-openness. We prove that the class of soft ωp-open sets is closed under soft union and do not form a soft topology, in general. We prove that soft ωp-open sets which are countable are soft open sets, and we prove that soft pre-open sets which are soft ω-open sets are soft ωp-open sets. In addition, we give a decomposition of soft ωp-open sets in terms of soft open sets and soft ω-dense sets. Moreover, we study the correspondence between the soft topology soft ωp-open sets in a soft topological space and its generated topological spaces, and vice versa. In addition to these, we define soft ωp-continuous functions as a new class of soft mappings which lies strictly between the classes of soft continuous functions and soft pre-continuous functions. We introduce several characterizations for soft pre-continuity and soft ωp-continuity. Finally, we study several relationships related to soft ωp-continuity.

Highlights

Introduction and PreliminariesIn this work, we follow the notions and terminologies of [1,2]
We prove that soft ωp-open sets which are countable are soft open sets, and we prove that soft pre-open sets which are soft ω-open sets are soft ωp-open sets
We study the correspondence between the soft topology soft ωp-open sets in a soft topological space and its generated topological spaces, and vise versa

Summary

Introduction and Preliminaries

We follow the notions and terminologies of [1,2]. TS and STS will denote topological space and soft topological space, respectively. K is called a soft pre-open set if there exists G ∈ τ such that K ⊆ G ⊆ Clτ(K). F is said to be a soft ωp-open set in (X, τ, A) if there exists G ∈ τ such that F ⊆ G ⊆ Clτω (F). Is soft anti-locally countable and F ∈ τω, by Theorem 14 of [2], Clτω (F) = Clτ(F). 1. (a) =⇒ (b): Suppose that fpu is soft pre-continuous and let M ∈ SS(X, A). Suppose that fpu : (X, τ( ), A) −→ (Y, τ(א), B) is soft pre-continuous. 1. (a) =⇒ (b): Suppose that fpu is soft ωp-continuous and let G ∈ τω.

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Conclusions