JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 81, 497-506 (1981) Fuzzy Hausdotff Topological Spaces REKHA SRIVASTAVA,~. N. LAL, AND ARUN K. SRIVASTAVA Department of Mathematics, Faculty of Science, Banaras Hindu University, Varanasi 221005, India Submitted by 15. A. Zadeh We introduce the notion of a fuzzy Hausdorff topological space and make a few observations to establish the appropriateness of this notion. 1. INTRODUCTION We introduce here the notion of a fuzzy Hausdorff topological space. The definition is natural and its appropriateness is established by showing that the theory of fuzzy Hausdorff topological spaces is largely parallel to its counterpart in general topology. In doing so, we have to slightly modify Wong’s definition [3] of “a fuzzy point belonging to a fuzzy set.” Fortunately, this modification does not disturb any existing result. Several equivalent forms of fuzzy Hausdorffness are given. A particularly pleasing observation relates to the two functors 3: TOP -+ FTOP and i: FTOP + TOP of Lowen [2], where TOP and FTOP respectively denote the categories of topological spaces and fuzzy topological spaces. We note that both these functors preserve Hausdorffness and (3 also reflects it. 2. PRELIMINARIES In this section we give several definitions (mostly known; see,for example, Wong [ 31) and some of their immediate consequences relevant to this paper. If A is a fuzzy set in X, its membership function is denoted bypu,. The membership function of an ordinary set is its characteristic function. DEFINITION 2.1. (13 1) A fuzzy point p in a set X is a fuzzy set in X given by P,(X) = t for x=x, (O<t< 1) and &4x> = 0 for x#x,. 497 0022.241X/81/060491-10902.oo/o Copyright 0 I90 I by Academic Press. Inc. All rights of reproduction in any form n%ervcd.