In the present paper, a characterization of the intuitionistic fuzzy sets, the interval-valued intuitionistic fuzzy sets and their set-operations are given. By making use of these characterizations, the relationships between the interval-valued intuitionistic fuzzy topology and four fuzzy topologies associated with it are studied. For this reason, some subclasses of the family of interval-valued intuitionistic fuzzy topologies on a set that we call pre-suitable and suitable are introduced. Furthermore, the concepts of homeomorphism functions and compactness in the framework of interval-valued intuitionistic fuzzy topological spaces are introduced and studied. Keywords: Interval-valued intuitionistic fuzzy set, interval-valued intuitionistic fuzzy topology, fuzzy topology, homeomorphism, compactness
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