sets and systems ELSEVIER Fuzzy Sets and Systems 90 (1997) 83-88 On several types of compactness in smooth topological spaces Mustafa Demirci Department of Mathematics, Faculty of Sciences and Arts, Abant izzet Baysal University, 14280-Bolu, Turkey Received June 1995; revised April 1996 Abstract The smooth closure and smooth interior of a fuzzy set w.r.t. a smooth topology were defined by Gayyar et al. (1994), and some relations between a few types of compactness were established in the presence of strong restrictions. In this paper, by constructing new definitions of smooth closure and smooth interior which have more desirable properties than those of Gayyar et al. (1994), we prove that several hypothesis in the results of Gayyar et al. (1994) can be weakened and show that the relations which hold between various types of compactness in fuzzy topological spaces in Chang’s sense (CFTS for short) (Di Concilio and Gerla, 1984; Haydar Es, 1987) can be extended to smooth topological spaces. 0 1997 Published by Elsevier Science B.V. Keywords: Fuzzy sets; Topology; Smooth almost compactness; Smooth near compactness 1. Introduction In 1985, Sostak [9] defined a fuzzy topology on a set X as a mapping r :I’ -+ I satisfying some natural axioms, where 1’ denotes the family of all fuzzy subsets of X, and presented the fundamental concepts of such fuzzy topological spaces. In 1992, the same structure was rediscovered by Chattopadyay et al. [2]. They call the mapping z : Ix -+ I “a gradation of openness on x”. In the same year, Ramadan [S] gave a similar definition of a fuzzy topology in Sostak’s sense under the name of “smooth topological spaces” (s.t.s.), replacing I = [0, l] by possibly more general lattices. smooth closure and smooth interior defined there do not have such nice properties as the closure and interior operators in a CFTS [7]. In this paper, we give a new definition of smooth closure and smooth interior of a fuzzy set in a s.t.s. which have almost all the properties of the corres- ponding operators in a CFTS. As a consequence of these definitions we reduce the additional hypotheses in the results of [4] and generalize sev- eral properties of the compactness’s types in [3,5] for s.t.s. 2. Preliminaries Degrizations of compactness and the types of compactness were introduced and studied in [6, lo]. A different approach for the compactness types was taken up in [4], however the results obtained include additional conditions since the Smooth topology and smooth topological con- cepts were introduced in [S] in terms of lattices L and L’, both of which were taken to be I = zyxwvutsrqpo [0, zyxwvutsr 11. For simplicity, we use only the symbol I instead of 0165-0114/97/$17.00 a 1997 Published by Elsevier Science B.V. All rights reserved PII SO165-01 14(96)00121-2