C-scattered fuzzy topological spaces

K e y w o r d s T o p o l o g y , C-scattered and scattered topological spaces. ('ompactness, Fuzzy perfect maps, S-paracompactness, S*-paracompactness. Fuzzy paracompactness. *-Fuzzy paracompactness. 1. I N T R O D U C T I O N The concept of C-sca t te red topological space has been defined by Telg{,rsky [1]. A space X is said to be C-sca t te red if each of its nonempty closed subspaces contains a compact set with n o n e m p t y relative interior. Then, it is a s imultaneous generalization of scat tered and of locally compac t spaces, and it is interesting because this class has good properties in relation with the p roduc t of pa racompac t spaces. See the papers of TelgArsky [1 4], Ormosadze [5], Dodon and ( ' oban [6,7], Friedler, Mar t in and Will iams [8], Nogura [9], Yaj ima [10] and the author [11 13]. In this paper, we define for fuzzy topological spaces in the Chang ' s sense, the not ion of C-sca t te red (and scat tered) spaces, and obta in various propert ies of these spaces. In p,trticular, the relationship between C-sca t te red spaces and their fuzzy extensions is studied, it is proved tha t the image by a fuzzy perfect map of all C-scat tered fuzzy topological space, is also C-scat tered, and that , in the reahn of pa racompac t fuzzy topological spaces (in the various senses), the C-sca t te red spaces verify tha t its product by other fuzzy spaces is also pa racompac t thzzy. Tho s tandard definitions on fuzzy topolooy are in [14.15], moreover, in this 1)aper we will use the following known notions. DEFINITION 1. (See [1@) Let (X, r ) be a topolog~ca] space and ~(r) be the set of all lower semicontinuous functions from (X, r) to the unit interval equipped uqth the usuM topoloKv, then (X. ~(r)) is called the weakly induced fuzzy topological space by (X, 7). 0893-9659/00/$ see front matter (~) 2000 Elsevier Science Ltd. All rights reserved Typeset by A~4S-rI~bX PlI: SI}893-9659(00)00136-1