Compactness and connectedness via the class of soft somewhat open sets

Tareq M. Al-shami; Abdelwaheb Mhemdi; Radwan Abu-Gdairi; Mohammed E. El-Shafei; Tareq M. Al-shami; Abdelwaheb Mhemdi; Radwan Abu-Gdairi; Mohammed E. El-Shafei

doi:10.3934/math.2023040

AIMS Mathematics

2023, Volume 8, Issue 1: 815-840. doi: 10.3934/math.2023040

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Compactness and connectedness via the class of soft somewhat open sets

1.
Department of Mathematics, Sana'a University, Sana'a, Yemen
2.
Department of Mathematics, College of Sciences and Humanities in Aflaj, Prince Sattam bin Abdulaziz University, Riyadh, Saudi Arabia
3.
Department of Mathematics, Faculty of Science, Zarqa University, P.O. Box 13110 Zarqa, Jordan
4.
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt

Received: 05 September 2022 Revised: 24 September 2022 Accepted: 28 September 2022 Published: 12 October 2022
MSC : 54A40, 03E72, 54D05, 54D20, 54D30

This paper is devoted to study the concepts of compactness, Lindelöfness and connectedness via the class of soft somewhat open sets which represents one of the generalizations of soft open sets. Beside investigation the main properties of these concepts, it is demonstrated, with the help of examples, that some properties of their counterparts via soft open sets are invalid. Also, the relationships between these concepts and their counterparts defined in classical topology (which is studied herein under the name of parametric topology) are discussed in detail. Moreover, we provide the sufficient conditions that guarantee the equivalence between them. In this regard, it is proved that all introduced types of soft compact and Lindelöf spaces are transmitted to all parametric topologies without imposing any conditions, whereas the converse holds true under the conditions of a full soft topology and a finite (countable) set of parameters. These characterizations represent a unique behavior of these spaces compared to the other types defined by celebrated generalizations of soft open sets. Also, there is no relationship associating soft $ sw $-connectedness with its counterparts via parametric topologies. We successfully describe soft $ sw $-disconnectedness using soft open sets instead of soft $ sw $-open sets and consequently prove that the concepts of soft $ sw $-connected and soft hyperconnected spaces are identical. In conclusion, the obtained results show that the framework given in this manuscript enriches and generalizes the previous works, and has a good application prospect.
- soft somewhat open set,
- soft compact,
- soft Lindelöf,
- soft connected,
- soft topology,
- parametric topology
Citation: Tareq M. Al-shami, Abdelwaheb Mhemdi, Radwan Abu-Gdairi, Mohammed E. El-Shafei. Compactness and connectedness via the class of soft somewhat open sets[J]. AIMS Mathematics, 2023, 8(1): 815-840. doi: 10.3934/math.2023040

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Abstract

This paper is devoted to study the concepts of compactness, Lindelöfness and connectedness via the class of soft somewhat open sets which represents one of the generalizations of soft open sets. Beside investigation the main properties of these concepts, it is demonstrated, with the help of examples, that some properties of their counterparts via soft open sets are invalid. Also, the relationships between these concepts and their counterparts defined in classical topology (which is studied herein under the name of parametric topology) are discussed in detail. Moreover, we provide the sufficient conditions that guarantee the equivalence between them. In this regard, it is proved that all introduced types of soft compact and Lindelöf spaces are transmitted to all parametric topologies without imposing any conditions, whereas the converse holds true under the conditions of a full soft topology and a finite (countable) set of parameters. These characterizations represent a unique behavior of these spaces compared to the other types defined by celebrated generalizations of soft open sets. Also, there is no relationship associating soft $ sw $-connectedness with its counterparts via parametric topologies. We successfully describe soft $ sw $-disconnectedness using soft open sets instead of soft $ sw $-open sets and consequently prove that the concepts of soft $ sw $-connected and soft hyperconnected spaces are identical. In conclusion, the obtained results show that the framework given in this manuscript enriches and generalizes the previous works, and has a good application prospect.

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