Abstract
We generally follow the terminology of Azad ( J. Math. Anal. Appl., 82 1981, 14–32) and Ming ( J. Math. Anal. Appl., 76 1980, 571–599). In addition to the fundamental concepts for fuzzy sets, we emphasize the usefulness of the concepts of fuzzy point-fuzzy elementhood. For fuzzy sets A and B a new characterization is given for relations A ⊂ B and A = B. This knowledge permits us to combine the two definitions of fuzzy point-fuzzy elementhood. In the third section some results are given concerning various special types of fuzzy sets in fuzzy topological spaces, and the fuzzy semi-regular, fuzzy regular spaces defined by Azad. In the last section, the definitions of H. almost continuous, W. almost open functions Urysohn space which are defined by Hussain, Wilansky, and Noiri, respectively, are extended to fuzzy sets. Furthermore some results are obtained in the functions of the fuzzy topological spaces defined by Azad and those are defined here.
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