JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 136, 124-130 (1988) On Fuzzy T, -Topological Spaces REKHA SRIVASTAVA,* S. N. LAL, AND ARUN K. SRIVASTAVA Department of Mathematics, Banaras Hindu Unicersity, Vuranasi-221 005, (U. P.), Indiu Submitted by L. A. Zadeh Received October 10. 1983 1. INTRODUCTION For a topological space (X, T), the following two statements are equivalent and define Hausdorffness: (i) Vx, y E X, x # y, 3U, VE T with XE.U, ye V, and Un V= 0 and (ii) A,, the diagonal of X, is closed in Xx X. The former property was used by Srivastava et al. [6] to define a fuzzy Hausdorff space while the latter property was used by Hutton [ 1] to define fuzzy Hausdorffness. Fortunately, in the case of fuzzy topological spaces also, the two ways of defining fuzzy Hausdorffness turn out to be equivalent (cf. Srivastava et al. [6]). Now recall that for a topological space (X, T) the following two statements are equivalent and define a T, space:(a)Vx,yEX,x#y,3U, VETsuchthatxEU,x4VandyEV,y~U and (b) {x}, VXE X, is closed in X. Srivastava et al. [7] earlier have used (a) to define fuzzy T,-ness while a case for using (b) to define a fuzzy T,- topological space is made out in the present note. However, this time the two ways of defining a fuzzy T,-space do not turn out to be equivalent. There are other definitions of a fuzzy T,-space also. In this note, we present a few observations bringing out the various relations among these different definitions of fuzzy T,-ness. 2. PRELIMINARIES A fuzzy topology r on a set X is a collection of fuzzy subsets in X which is closed under arbitrary suprema and finite infima and which contains both 0 and X (Wong [S]). Lowen [2], however, has called r a fuzzy topology if, in addition to the above, it also contains all constant fuzzy sets. The term “fuzzy topological space” will be abbreviated as fts. * Financial support through a CSIR Senior Research Fellowship is gratefully acknowledged. 124 0022-247X/88 $3.00 CopyrIght ( 1988 by Academic Press. Inc All rights of reproductmn m any form reserved