Fuzzy set theory provides us with a framework which is wider than that of classical set theory; various mathematical structures, whose features emphasize the effects of ordered structure, can be developed on the theory. Fuzzy topology is one such branch combining ordered structure with topological structure. This branch of mathematics emerged from the background - processing fuzziness, with locale theory, proposed from the angle of pure mathematics by the great French mathematician Ehresmann. It comprises, surpisingly, the two most active aspects of topology on the lattice, which affect each other. This book is a monograph which systematically reflects the up-to-date state of fuzzy topology. It emphasizes the so-called "pointed approach" and the effects of stratification structure appearing in fuzzy sets. The monograph can serve as a reference book for mathematicians, researchers and graduate students working in this branch of mathematics. After an appropriate rearrangement of the chapters and sections, it can also be used as a text for advanced undergraduates.