We have carefully put this book for those interested in the different branches of fuzzy mathematics that are used in a variety of applications in other scientific fields such as engineering, computer science, biology, geography and others. In recent years, there is an increased interest in the concept of fuzzy sets that led to the concept of fuzzy logic which is used in several areas, such as control-theory, game-theory, economics, statistics, data management, artificial intelligence, automated reasoning etc. What distinguishes this book is that it contains many examples not available before, and characterized by their lengthy accounts. The origin of this difficulty is the nature of fuzzy sets provided in 1965 by Lotfi Zadeh which are more complicated than the ordinary set. Our system in this work depends on using the idea of the generalized closed sets studied by Levin, in 1970, which I identified it on the fuzzy sitting on different types. It is a powerful mean to describe the topological spaces that meet weak separation axioms applicable in computer science. Throughout this work, we elucidate new born results, theories and fuzzy continuous mappings in [0,1]-topological spaces.