Soft weakly connected sets and soft weakly connected components

Samer Al-Ghour; Hanan Al-Saadi; Samer Al-Ghour; Hanan Al-Saadi

doi:10.3934/math.2024077

AIMS Mathematics

2024, Volume 9, Issue 1: 1562-1575. doi: 10.3934/math.2024077

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Research article

Soft weakly connected sets and soft weakly connected components

Samer Al-Ghour ^{1
,
,},
Hanan Al-Saadi ^{2
,
,}

1.
Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid 22110, Jordan
2.
Department of Mathematics, Faculty of Applied Sciences, Umm Al-Qura University, Makkah 24225, Saudi Arabia

Received: 07 August 2023 Revised: 28 November 2023 Accepted: 29 November 2023 Published: 12 December 2023
MSC : 54A40, 54D05

Although the concept of connectedness may seem simple, it holds profound implications for topology and its applications. The concept of connectedness serves as a fundamental component in the Intermediate Value Theorem. Connectedness is significant in various applications, including geographic information systems, population modeling and robotics motion planning. Furthermore, connectedness plays a crucial role in distinguishing between different topological spaces. In this paper, we define soft weakly connected sets as a new class of soft sets that strictly contains the class of soft connected sets. We characterize this new class of sets by several methods. We explore various results related to soft subsets, supersets, unions, intersections and subspaces within the context of soft weakly connected sets. Additionally, we provide characterizations for soft weakly connected sets classified as soft pre-open, semi-open or $ \alpha $-open sets. Furthermore, we introduce the concept of a soft weakly connected component as follows: Given a soft point $ a_{x} $ in a soft topological space $ \left(X, \Delta, A\right) $, we define the soft weakly component of $ \left(X, \Delta, A\right) $ determined by $ a_{x} $ as the largest soft weakly connected set, with respect to the soft inclusion ($ \widetilde{\subseteq } $) relation, that contains $ a_{x} $. We demonstrate that the family of soft weakly components within a soft topological space comprises soft closed sets, forming a soft partition of the space. Lastly, we establish that soft weak connectedness is preserved under soft $ \alpha $-continuity.
- weakly connected sets,
- soft connectedness,
- soft $ \alpha $-open sets,
- soft connected components
Citation: Samer Al-Ghour, Hanan Al-Saadi. Soft weakly connected sets and soft weakly connected components[J]. AIMS Mathematics, 2024, 9(1): 1562-1575. doi: 10.3934/math.2024077

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Abstract

Although the concept of connectedness may seem simple, it holds profound implications for topology and its applications. The concept of connectedness serves as a fundamental component in the Intermediate Value Theorem. Connectedness is significant in various applications, including geographic information systems, population modeling and robotics motion planning. Furthermore, connectedness plays a crucial role in distinguishing between different topological spaces. In this paper, we define soft weakly connected sets as a new class of soft sets that strictly contains the class of soft connected sets. We characterize this new class of sets by several methods. We explore various results related to soft subsets, supersets, unions, intersections and subspaces within the context of soft weakly connected sets. Additionally, we provide characterizations for soft weakly connected sets classified as soft pre-open, semi-open or $ \alpha $-open sets. Furthermore, we introduce the concept of a soft weakly connected component as follows: Given a soft point $ a_{x} $ in a soft topological space $ \left(X, \Delta, A\right) $, we define the soft weakly component of $ \left(X, \Delta, A\right) $ determined by $ a_{x} $ as the largest soft weakly connected set, with respect to the soft inclusion ($ \widetilde{\subseteq } $) relation, that contains $ a_{x} $. We demonstrate that the family of soft weakly components within a soft topological space comprises soft closed sets, forming a soft partition of the space. Lastly, we establish that soft weak connectedness is preserved under soft $ \alpha $-continuity.

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