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Mathematical Methods in Science and Mechanics On Some Paracompactness-type Properties of Fuzzy Topological Spaces FRANCISCO GALLEGO LUPIÁÑEZ Dept. Geometry & Topology, and IMI Fac. Mathematics Univ. Complutense of Madrid 28040 Madrid, SPAIN e-mail: [email protected] October 15, 2014 Abstract are good extensions of those de…ned by A.V. Arhangel’skiii, and obtain several results about The aim of this paper is to study some them. paracompactness-type properties for fuzzy topoFirst, we give some de…nitions: logical spaces. we prove that these properties are good extensions of others de…ned by A.V.Arkhangel’skii (and studied by S.A. Peregu- De…nition 1 Let be a fuzzy set in a fuzzy dov) and obtain several results about them. topological space (X; ). We will say that Keywords: Fuzzy sets; Topology; paracomis fuzzy a -paracompact (respectively, apactness; fuzzy perfect maps. paracompact) if for each open cover (in Lowen’s AMS Mathematics Subject Classi…cation sense) U of and for each 2 (0; 1], there (2010): 54A40, 03E72 exits an open re…nement V of U which covers (in Lowen’s sense) and every in…nite subfamily of V contains an in…nite pairwise disjoint subfamily (respectively, contains a disjoint pair 1 Introduction of elements). We will say that (X; ) is fuzzy On 1971-1972, A.V. Arhangel’kii introduced a -paracompact (respectively, a-paracompact) if some new paracompactness-type properties: a- each constant fuzzy set in X is fuzzy a paracompactness, a -paracompactness and b- paracompact (respectively, a-paracompact). paracompactness. Later, S.A. Peregudov did some research on these concepts [9,10]. De…nition 2 Let be a fuzzy set in a fuzzy is In this paper, we de…ne fuzzy extensions topological space (X; ). We will say that of these notions, prove that these properties fuzzy b-paracompact if for each open cover (in 1 ISBN: 978-960-474-396-4 238 Mathematical Methods in Science and Mechanics For each V 2 U , there exits GV 1 (r ; 1] 2 U such that V GV 1 (r ; 1]: Then B = f V ^ GV = V 2 U g and re…nes B. Now, for each x 2 X, there exits V 2 U such that V (x) = 1 and GV_ (x) › r . Thus, ( V ^ GV ) (x) r and ( V ^ GV ) (x) r Lowen’s sense) U of and for each 2 (0; 1], there exits an open re…nement V of U which covers (in Lowen’s sense) and every subfamily of V with the …nite intersection property, is …nite. We will say that (X; ) is fuzzy bparacompact if each constant fuzzy set in X is fuzzy b-paracompact. . V 2U For each B1 B such that B1 has the …De…nition 3 [4] Let (X; T ) be a topological space, !(T ) = f : (X; T ) ! I j is a lower nite intersection property, we have that also, fVj semi-continuos mapg:The fuzzy topological space / Vj ^ GV j 2 B1 g has the …nite intersection property, because, if for each f V1 ^ GV1 ; ::: Vn ^ (X; !(T )) is said induced by (X; T ). GVn g B1 is ( V1 ^ GV1 ) ^ ::: ^ ( Vn ^ GVn ) 6= 0, De…nition 4 [4] Let P be a property of topo- there exists x 2 X such that, for each i 2 logical spaces and P be a fuzzy extension of P. f1; :::ng is ( V i ^ GVi )(x) 6= 0 (then, x 2 Vi The concept P is a good extension of P if a and GVi (x) 6= 0): So, V1 \ ::: \ Vn 6= ?: topological space (X; T ) veri…es P if and only if From the hypothesis, fVj / Vj ^ GV j 2 B1 g (X; !(T )) veri…es P : is …nite, then B1 is …nite. Thus, (X, ) is fuzzy b-paracompact. De…nition 5 [7] A fuzzy topological space ((=) Let U [ ] be an open cover of (X; [ ]), (X; ) is a weakly inducement of an ordinary then U 0 = f U / U 2 Ug is an_ open fuzzy cover topological space (X; T ) if: r. For U 2 g is of (X; );and for each r 2 I; (a) T = [ ], where [ ] = fA X = A U 2U the original topology of : each 2 (0; r], there exits an open re…nement (b) every 2 is a lower semi-continuos map G of U 0 such that every subfamily with _ the …nite from (X; T ) into I = [0; 1]: intersection property is …nite, and G r , 2 which implies that Results _ G2G G r 1 for all 1 › : G2G Let U = fG 1 (r ; 1] / G 2 Gg, then U [ ] is an open re…nement of U (because, for each G 1 (r ; 1] 2 U there exits VG 2 U such that G , and G 1 (r ; 1] V ); and U ia VG _ G an open cover of X; because G r 1: Theorem 6 A topological space (X; [ ]) is bparacompact if and only if (X; ) is fuzzy bparacompact. Proof. ( =) ) _ For each r 2 I, let a family B such that G r: Let 2 (0; r], then G2G G2B For each U1 U such that U1 has the …nite intersection property, we have that fGj / Gj 1 (r ; 1] 2 U1 g has also the …nite intersection property (because, if for each fG1 1 (r ; 1] 1 1 ; :::; Gn (r ; 1] g U1 is G1 (r ; 1] \::: \ the family U =fG 1 (r ; 1] = G 2 Bg [ ] is an open cover of (X; [ ]), and from the hypothesis, it has an open re…nement U [ ] which is a cover of and every subfamily of U with the …nite intersection property is …nite. 2 ISBN: 978-960-474-396-4 239 Mathematical Methods in Science and Mechanics Gn 1 (r ; 1] 6= ?, then for each i = 1; ::; n ,there is VGi 2 U such that Gi VGi and there is x 2 X such that Gi (x) › r for all i = 1; :::; n. So, (G1 ^ ::: ^ Gn )(x) 6= 0). By the hypothesis, fGj / Gj 1 (r ; 1] 2 U1 g is …nite, then U1 is …nite. Thus, (X; [ ]) is b-paracompact. an in…nite pairwise disjoint subfamily (resp. contains a disjoint pair of elements). Thus, B1 has an in…nite pairwise disjoint subfamily (resp. has a disjoint pair of elements), because Vj1 \ Vj2 = ? implies that ( Vj ^ GVj1 ) ^ ( Vj ^ 1 2 GVj2 ) = Vj \Vj ^ (GVj1 ^ GVj2 ) = 0: 1 2 So, (X, ) is fuzzy a -paracompact (resp. aparacompact). ((=) Let U [ ] be an open cover of (X; [ ]), Remark 7 If (X; T ) is a topological space, open fuzzy cover then (X; T ) is b-paracompact if and only if then U 0 = f U / U 2 Ug is an_ (X; !(T )) is fuzzy b-paracompact. (i.e. fuzzy of (X; );and for each r 2 I; r. For U b-paracompactnesss is a good extension of bU 2U each 2 (0; r], there exists an open re…nement paracompactness). Proof. By the above proof (now it is also valid). G of U 0 such that every in…nite subfamily of G contains an in…nite pairwise disjoint subfamily (resp. contains a disjoint pair of elements), and _ G r . G2G Theorem 8 A topological space (X; [ ]) is a paracompact (resp. a-paracompact) if and only if (X; ) is fuzzy a -paracompact (resp .aparacompact). Proof. ( =) ) _ For each r 2 I, let a family B such that G r: Let 2 (0; r], then Analogously to the previous theorem, U = fG 1 (r ; 1] / G 2 Gg re…nes U and is an open cover of X: For each in…nite subfamily U1 of U we have that U1 = fGj 1 (r ; 1] = j 2 Jg for some in…nite set J. Then fGj =j 2 Jg is in…nite and contained in G: Thus, there exits an in…nite pairwise disjoint subfamily (resp. exists a disjoint pair of elements) of it, and also U1 has an in…nite pairwise disjoint subfamily (resp. has a disjoint pair of elements), because GVj1 ^ GVj2 = 0 implies that Gj11 (r ; 1]\ Gj21 (r ; 1] = ? (if Gj1 (x0 ); Gj2 (x0 ) 2 (r ; 1] for some x0 2 X; then (Gj1 ^ Gj2 )(x0 ) 6= 0; and this is contradictory). So, (X; [ ]) is a -paracompact (resp. aparacompact). G2B the family U =fG 1 (r ; 1] = G 2 Bg [ ] is an open cover of (X; [ ]), and by the hypothesis, it has an open re…nement U [ ] which is a cover of and every in…nite subfamily of U contains an in…nite pairwise disjoint subfamily (resp. contains a disjoint pair of elements). Analogously to the previous theorem, for each V 2 U , there is GV 1 (r ; 1] 2 U such that V GV 1 (r ; 1]:Then B = f V ^ GV = V 2 U g ; re…nes _ B, and ( V ^ GV ) (x) r . V 2U B , B1 = n For each in…nite osubfamily B1 Remark 9 If (X; T ) is a topological space, for some in…nite set J, then (X; T ) is a -paracompact (resp. Vj ^ GV j = j 2 J athen fVj / j 2 Jg U is in…nite and contains paracompact) if and only if (X; !(T )) is fuzzy a 3 ISBN: 978-960-474-396-4 240 Mathematical Methods in Science and Mechanics paracompact (resp.a-paracompact) (i.e. fuzzy a - 3 References paracompactness and fuzzy a-paracompactness are good extensions of a -paracompactness and [1] M.E.Abd El-Mosef, F.M.Zeyada, S.N.ElDeeb and I.M.Hanafy: "Good extensions of a-paracompactness). Proof. By the above proof (now it is also valid). paracompactness", Math. Japon. 37 (1992), 195-200. [2] H.Aygün, M.W.Warner and S.R.T.Kudri: "Completely induced L-fuzzy topological Proposition 10 All fuzzy paracompact and spaces", Fuzzy Sets Systems 103 (1999), Hausdor¤ fuzzy topological space is fuzzy a 513-523. paracompact. [3] S.R.T.Kudri and M.W.Warner:"L-fuzzy local compactness", Fuzzy Sets Systems 67 (1994), Proof. Firstly, fuzzy paracompactness and fuzzy Hausdor¤ ness are good extensions ([1,5]). 337-345. [4] R.Lowen: "A comparison of di¤erent comAlso, every Hausdor¤ paracompact space is a pactnesss notions for fuzzy topological spaces", paracompact ([9]), and …nally by last Remark, J.Math.Anal.Appl. 64 (1978), 446-454. fuzzy a -paracompactness is a good extension. [5] R.Lowen: "CompactHausdor¤ fuzzy topological spaces are topological", Topology Appl. 12 (1981), 65-74. [6] F.G.Lupiáñez: "Fuzzy perfect maps and Proposition 11 The properties fuzzy b- fuzzy paracompactness", Fuzzy Sets Systems 98 paracompactness, a-paracompactness and (1998), 137-140. a -paracompactness are preserved by preimages [7] M.K.Luo: "Paracompactness in fuzzy topoof fuzzy perfect maps in the sense of Ghosh (rep. logical spaces", J.Math.Anal.Appl.130 (1988), Srivastava and Lal, Christoph). 55-77. [8] H.W.Martin: "Weak induced fuzzy topoProof. It follows from above Remarks and [6,9]. logical spaces", J.Math.Anal.Appl. 78 (1980), 634-639. [9] S.A.Peregudov: "On the classi…cation of Proposition 12 Let f : (X; ) ! (Y; &) be paracompactness-type properties of topological a fuzzy perfect map in the Ghosh’s sense (resp. spaces", Vesnik Moskov Univ. Ser I Mat. Meh. Christoph or Srivastava and Lal) of a fuzzy lo28 no 6 (1973), 3-9 (Russian) (trasl.:Moscow cally compact b-paracompact space (X; ) onto Univ.Math. Bull. 28 no 6 (1974), 55-60. the fuzzy Hausdor¤ space (Y; &). Then, (Y; &) is [10] S.A.Peregudov: "Some remarks on bfuzzy b-paracompact. paracompact spaces", Mat.Zametki 22 no 4 Proof. We have that fuzzy locally compact- (1977), 485-494 (Russian) ness and fuzzy Hausdor¤ness are good extensions ([3,5]), and also fuzzy b-paracompactness (by the …rst Remark). Then, the result follows from [9, Th.5]. 4 ISBN: 978-960-474-396-4 241