[Paper Review] Soft Somewhat Continuous and Soft Somewhat Open Functions

Zanyar A. Ameen, Baravan A. Asaad+1 more|arXiv (Cornell University)|Dec 30, 2021

Fuzzy and Soft Set Theory20 citations

TL;DR

This paper introduces soft somewhat open sets using the soft interior operator and defines soft somewhat continuous and soft somewhat open functions as generalizations of soft semicontinuous and soft semi-open functions. The key contribution is establishing that soft somewhat homeomorphisms do not preserve standard soft topological properties like soft metrizability, compactness, or Hausdorffness, highlighting a fundamental distinction from classical soft homeomorphisms.

ABSTRACT

In this paper, we define a soft somewhat open set using the soft interior operator. We study main properties the class of soft somewhat open sets that is contained in the class soft somewhere dense sets. Then, we introduce the classes of soft somewhat continuous and soft somewhat open functions and soft somewhat homeomorphisms. Moreover, we study properties and characterizations of soft somewhat continuous and soft somewhat open functions. At last, we discuss topological invariants for soft somewhat homeomorphisms. Multiple examples are offered to clarify some invalid results.

Motivation & Objective

• To define and investigate soft somewhat open sets using the soft interior operator.
• To introduce and characterize soft somewhat continuous and soft somewhat open functions as intermediate concepts between soft semicontinuity and soft somewhere dense continuity.
• To define soft somewhat homeomorphisms and examine their properties in relation to topological invariance.
• To demonstrate that soft somewhat homeomorphisms do not preserve standard soft topological properties such as soft metrizability, compactness, or Hausdorffness.

Proposed method

• Define soft somewhat open sets as those whose soft interior is non-null in the codomain.
• Introduce soft somewhat continuous functions via preimages of soft open sets being soft somewhat open.
• Define soft somewhat open functions as those mapping non-null soft open sets to sets with non-null soft interior.
• Establish relationships between soft somewhat open sets and other generalized soft open sets, such as soft semiopen and soft β-open sets.
• Prove that soft somewhat open functions are weaker than soft semi-open functions but stronger than soft somewhere dense open functions.
• Use counterexamples to show that soft somewhat homeomorphisms do not preserve soft topological properties like metrizability, local compactness, or the T₀ separation axiom.

Experimental results

Research questions

• RQ1 How do soft somewhat open sets relate to other generalized soft open sets like soft semiopen and soft β-open sets?
• RQ2 What are the necessary and sufficient conditions for a function to be soft somewhat continuous or soft somewhat open?
• RQ3 Can soft somewhat homeomorphisms preserve standard soft topological properties such as soft compactness or soft Hausdorffness?
• RQ4 What are the implications of soft somewhat homeomorphisms for the invariance of soft topological structures?
• RQ5 How do soft somewhat open sets compare to soft somewhere dense sets in terms of inclusion and independence?

Key findings

• Soft somewhat open sets are contained within the class of soft somewhere dense sets but are independent of soft β-open sets.
• Soft somewhat continuous functions are weaker than soft semicontinuous functions but stronger than soft somewhere dense continuous functions.
• Soft somewhat open functions are weaker than soft semi-open functions but stronger than soft somewhere dense open functions.
• Soft somewhat homeomorphisms do not preserve soft metrizability, soft local compactness, or soft connectedness, as shown by counterexamples.
• The identity map from the standard soft topology on ℝ to the soft Sorgenfrey line is a soft somewhat homeomorphism, yet the codomain lacks soft metrizability and soft local compactness.
• Soft somewhat homeomorphisms also fail to preserve separation axioms, as demonstrated by a soft T₀ space that is not soft T₁ despite being the image under a soft somewhat homeomorphism.