New kind of vector space via Ş-Proximit structure

Abstract

This paper describes the basic aim of Ş -proximit space for non-empty sets. The relationship between approach space and Ş-proximit space has been clarified. This paper demonstrates that all metric spaces are Ş-proximit spaces. But the opposite is not always true. We define SA-contraction and present various features and the results. In addition, we define new Ş-proximit normed space. We demonstrate that any proximit normed space is normed. But the opposite is not always true. We define the concepts of the Ş -proximit semigroup, Ş-proximit group, and Ş-proximit vector space via instances of problem-solving techniques. We also discussed the definitions of Cauchy sequence, cluster point, Ş -convergent sequence, and Ş -complete.