Pu and Liu's Q-theory is combined with Lowen's goodness criterion for fuzzy extensions to provide a framework for fuzzifying topology. This framework is used for the study of fuzzy countability properties and for the fuzzification of classical sequentiality. In extending classical notions to fuzzy theory care is taken to ensure that they are a special case of the emerging fuzzy concepts. An examination of convergence in the sense of Pu and Liu in special fuzzy topological spaces demonstrates the advantage of Chang's definition of fuzzy topology, which is therefore adopted. A new criterion (called excellence) for the suitability of the fuzzy extensions of classical topological properties is introduced. In addition to passing Lowen's goodness test, an excellent property is expected to behave, under fuzzy extensions of induction and coinduction, in a way resembling that of the original classical property under these constructions. Fuzzy second countability, quasi-first countability and fuzzy sequentiality are found to be excellent extensions of classical second countability, first countability and sequentiality respectively.