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The concepts of fuzzifying γ-open sets and fuzzifying γ-closed sets are studied and some interesting results (Theorems 5.4 and 5.5) are obtained. Also, the concept of fuzzifying γ-continuity are introduced and some important characterizations (Theorem 6.1) are obtained. Furthermore, some compositions of fuzzifying γ-continuity and fuzzifying continuity are presented (Theorem 6.2). @2002, International Society for Mathematical Sciences
The concepts of fuzzy pre-continuity and fuzzy cpre-continuity are introduced and studied in fuzzifying topology essentially in order to give decompositions of fuzzy continuity.
@ 2001 Elsevier Science B.V.
In the present paper we introduce R0- and R1-separation axioms in fuzzifying topology and study their relations with T1- and T2-separation axioms, respectively. Furthermore, we introduce and study semi-T0-, semi-R0-, semi-T1-, semi-R1-, semi-T2 (semi-Hausdorff)-, semi-T3 (semi-regularity)- and semi-T4 (semi-normality)-separation axioms in fuzzifying topology and
give some of their characterizations as well as the relations of these axioms and other separation axioms in fuzzifying topology introduced by Shen, Fuzzy Sets and Systems 57 (1993) 111–123.
@ 2001 Elsevier Science B.V.
The concept of semi-open sets is extended in fuzzifying topology in two various types one of them is stronger than the other, but the converse may not be true. These concepts and two types of semi-continuity induced by them are introduced and studied in fuzzifying topology. Furthermore, four types of irresolute functions are considered between fuzzifying topological spaces.
The concepts of α-continuity and cα-continuity are considered and studied in fuzzifying topology and by making use of these concepts, some decompositions of fuzzy continuity are introduced. It is proved that the family of all α-sets in
fuzzifying topology may not be a fuzzifying topology.
@ 2000 Elsevier Science B.V.
The concepts of β-continuity and D(c, β)-continuity are considered and studied fuzzifying continuity in fuzzifying topology and by making use of these concepts, some decompositions of fuzzy continuity are introduced.
@1999, International Fuzzy Mathematics Institute, Los Angeles.
In this paper, a preuniform structure based on way below relation (or L-fuzzifying preuniform structure) was defined and their properties were studied. Also, the concept of interior and closure operators in the L-fuzzifying setting were established. Furthermore, the relation between L-fuzzifying preuniform and L-fuzzifying topological spaces were explained. Finally, the continuity of L-fuzzifying preuniform spaces was studied.
The concepts of fuzzy semi-continuity and fuzzy csemi- continuity are introduced and studied essentially in order to decompose fuzzy continuity in fuzzifying topology. Furthermore, the concept of csemi-neighborhood system is presented and by making use of it a fuzzifying topology is introduced and so Comparisions of motivated types of continuity are pointed out.
Papilionoideae is the most species-rich subfamily of Fabaceae. Within Papilionoideae, the “core genistoids” comprise many genera of the tribes Podalyrieae, Thermopsideae, Euchresteae, Crotalarieae, Genisteae, and part of Sophoreae. Seed macro- and micro-morphological characters and seed storage proteins of 12 Egyptian species belonging to three tribes of the core genistoides (Sophoreae, Crotalarieae, and Genisteae) were examined using light microscopy, scanning electron microscopy, and SDS-PAGE electrophoresis. Seed shape, size, surface topography, hilum characteristics, epidermal cell patterns, periclinal cell walls, and anticlinal cell boundaries are described here. Seed protein profiles show variable banding patterns with molecular weights ranging from approximately 5 to 270 kDa. Multivariate analyses based on seed morphology and storage proteins elucidate the phenetic relationships among the investigated species at both the tribal and intrageneric levels.
The taxonomic and phylogenetic implications of multivariate analyses are compared with previous and current systematic treatments of the genera within the three tribes. Our results confirm the high taxonomic importance of seed morphology in distinguishing among the investigated species. A key to the species using seed macro- and micro-morphological characters is presented.