eng Science and Education Publishing American Journal of Applied Mathematics and Statistics 2328-7292 2022-08-31 10 2 65 68 10.12691/ajams-10-2-4 AJAMS20221024 article Michael Gr. Voskoglou mvoskoglou@gmail.com 1 Department of Mathematical Sciences, Graduate Technological Educational Institute of Western Greece, Patras, Greece The present paper reviews the process that led from Zadeh¡¯s fuzziness to Smarandache¡¯s neutrosophy and discusses the future perspectives of the corresponding theories. Starting from the definitions of fuzzy set and of Atanassov¡¯s intuitionistic fuzzy set, it proceeds to a detailed study of the concept of neutrosophic set, which takes in account the existing in real life indeterminacy. The basic operations of complement, union and intersection between neutrosophic sets are defined and the classical notion of topological space is extended to the notion of neutrosophic topological space. It is further shown that the fundamental concepts of convergence, continuity, compact topological space and Hausdorff topological space can be naturally extended to neutrosophic topological spaces. http://pubs.sciepub.com/ajams/10/2/4/ajams-10-2-4.pdf Fuzzy Set (FS) Fuzzy Logic (FL) Intuitionistic Fuzzy Set (IFS) indeterminacy Neutrosophic Set (NS) Neutrosophic Topological Space (NTS)