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A New Approach to Vague Soft Bi-Topological Spaces

Arif Mehmood1, Saleem Abdullah2, Choonkil Park3,*
1 Department of Mathematics & Statistics, Riphah International University, Islamabad, 44000, Pakistan
2 Department of Mathematics, Abdul Wali Khan University, Mardan, 23200, Pakistan
3 Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul, 04763, Korea
* Corresponding Author: Choonkil Park. Email:
(This article belongs to this Special Issue: Advances in Neutrosophic and Plithogenic Sets for Engineering and Sciences: Theory, Models, and Applications (ANPSESTMA))

Computer Modeling in Engineering & Sciences 2022, 131(1), 411-428.

Received 15 April 2021; Accepted 27 September 2021; Issue published 24 January 2022


Fuzzy soft topology considers only membership value. It has nothing to do with the non-membership value. So an extension was needed in this direction. Vague soft topology addresses both membership and non-membership values simultaneously. Sometimes vague soft topology (single structure) is unable to address some complex structures. So an extension to vague soft bi-topology (double structure) was needed in this direction. To make this situation more meaningful, a new concept of vague soft bi-topological space is introduced and its structural characteristics are attempted with a new definition. In this article, new concept of vague soft bi-topological space (VSBTS) is initiated and its structural behaviors are attempted. This approach is based on generalized vague soft open sets, known as vague soft open sets. An ample of examples are given to understand the structures. For the non-validity of some results, counter examples are provided to pay the price. Pair-wise vague soft open and pair-wise vague soft close sets are also addressed with examples in (VSBTS). Vague soft separation axioms are initiated in (VSBTS) concerning soft points of the space. Other separation axioms are also addressed relative to soft points of the space. Vague soft bi-topological properties are studied with the application of vague soft open sets with respect to soft points of the spaces. The characterization of vague soft close as well as vague soft open sets, characteristics of Bolzano Weirstrass property, vague soft compactness and its marriage with sequences, interconnection between sequential compactness and countable compactness in (VSBTS) with respect to soft open sets are addressed.


Vague soft set; vague soft point; vague soft bi-topological space; vague soft β-open set and vague soft β-separation axioms

Cite This Article

Mehmood, A., Abdullah, S., Park, C. (2022). A New Approach to Vague Soft Bi-Topological Spaces. CMES-Computer Modeling in Engineering & Sciences, 131(1), 411–428.


This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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