An evaluative overview of the definitions of Cantor set, Fuzzy set and Extension set and their applications, limitation and characteristics

This article will introduce Cantor sets and their importance in mathematics, especially in areas such as topology and number theory. Also, an overview of how Fuzzy Sets and Extension sets can be useful in areas such as uncertainty handling, decision science, and more. The paper will be organized around the theoretical foundations of Cantor sets, Fuzzy Sets, and Extension sets. The basic principles of construction are presented, as well as important features derived from the construction methods, revealing the advantages as well as the disadvantages that each type of feature exhibits in its application. It will also analyze their performances and connections in different application scenarios. The review will help readers quickly understand an overview of set theory and the fundamentals of the three set theories mentioned above. This paper will introduce the applications of these theories in modern mathematics and computation, engineering, and finance, point out the limitations of the different set theories, respectively, and then provide possible directions for future research.