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FUZZY TOPOLOGICAL PROPERTIES OF FUZZY POINTS AND ITS APPLICATIONS

H. MAKI, T. FUKUTAKE, M. KOJIMA, F. TAMARI, T. KONO, S. NITA, T. HAYASHI, S. HAMADA, H. KUWANO

2012
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Scientiae mathematicae Japonicae
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The present paper studies new properties of the concept of fuzzy points in the sense of Pu Pao-Ming and Liu Ying-Mi時(Defini tion 2.3 , Theorem 3.1). We first prove that, for an arbitrary Chang う s fuzzy topological space (Y, Ty), every fuzzy set入in Y with入# Oy is de composed by at most three fuzzy sets:入=入1V入 2 V ,\3 with入包 〈入3 二 Oy for each distinct integers i and j(l三i,j三3) (Theorem 2.10); moreover 入1 is fuzzy preopen in (Y, Ty) (Theorem 2.9(i)). Especially, if Ty is a fuzzy topology, sayσ f
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... cf. Example II in Section 3) which is induced from an ordinary topology σof Y, then every fuzzy set入in Y is decomposed by at most two fuzzy sets:入ニ入1V入 2 with入1八入 2 = Oy (Corollary 3.7(i)); and入1 is fuzzy preopen in the fuzzy topological space (Y,σ f ) (Corollary 3. 7(ii) ). Moreover, every fuzzy point (in the sense of Pu Pao-Ming and Lin Yi昭-Ming) is fuzzy open or fuzzy nowhere dense in (Y, σ f ) (Theorem 3.1). As applications, the results are applied to the case where Y := Z 2 and σニκ 2 (=the Kl凶imsky topology), i.e., (Y.,σ) = (Z 2 , r,; 2 ) is the digital plane. So, every dig出l image(手0) on Z 2 is decomposed by at most two digital images and they have such fuzzy topological properties (Theorem 3.9 in Section 3(III-5)). 1 Introduction and preliminaries In 1965 ぅ L.A. Zadeh [29] introduced and investigated the fundamental notion of fuzzy sets and fuzzy sets operations. Subsequently several authors applied various basic concepts from general topol ogy to fuzzy sets and developed the theory of Fuzzy topological spaces. In 1968, C.L. Cha時 [ 6 ] introduced and investigated the concept of fuzzy topolog ical spaces ( cf. Definition 1.1 below). In 197 4 う K.K. Wo時 [27, Definition 3.1] introduced and investigated the notion of fuzzy po肌ts (cf.[27, Theorem 3.1 an p.319]). In 1980, Pu Pao-Mi略and Liu Ying-Ming [24, Definition 2.1] redefined the concept of fuzzy points; it takes a crisp singleton, equivalentely, an ordinary point as a special case. In the present paper, we adopte and u日e the definition

doi:10.32219/isms.75.2_235
fatcat:vujqs432u5hklmclvbx3zsr6yu