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Báo cáo khoa học: "Một số tính chất của họ CF và cs-ánh xạ phủ compac" doc

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CF CS (1) CF CF,HCF HCP, CP CF W HCP k cs cs ¨o k k k 1. T 1 X P X P x ∈ X U x P ∈ P x ∈ P ⊂ U P k K U K X F ⊂ P K ⊂ ∪F ⊂ U X P X 1 2 P K X K P P cs A A P P CF K X {K ∩ P : P ∈ P} P = {P α : α ∈ Λ} HCF Γ ⊂ Λ A α ⊂ P α α ∈ Γ {A α : α ∈ Γ} CF X P = {P α : α ∈ Λ} X P CP I ⊂ Λ  {P α : α ∈ I} =  {P α : α ∈ I}. P HCP I ⊂ Λ Q α ⊂ P α , α ∈ I  {Q α : α ∈ I} =  {Q α : α ∈ I}. P W HCP I ⊂ Λ x α ∈ P α , α ∈ I  {x α : α ∈ I} =  {x α : α ∈ I}. P x ∈ X U x x U x P P x ∈ X ⇒ cs ⇒ HCF ⇒ CF ⇒ CF ⇒ HCP ⇒ CP WHCP X P X X P U ⊂ X X U ∩ P P P ∈ P X A ⊂ X A X A A X X k X f : X → Y f f(A) Y A X f U Y f −1 (U) X f K Y L X f(L) = K f s f −1 (y) X y ∈ Y f cs f −1 (K) X K Y f ¨o f −1 (y) ¨o X y ∈ Y f ¨o f −1 (L) ¨o X ¨o L Y cs ⇒ s ¨o ⇒ ¨o ⇒ ⇒ 2. W HCP CF X k P X P P P cs cs P ⇒ P K X x ∈ K P U x x U x P {U x : x ∈ K} K {U x 1 , U x 2 , . . . , U x n } {U x : x ∈ K} K ⊂  n i=1 U x i U x i P i = 1, 2, . . . , n K P P ⇒ ⇒ ⇒ P x ∈ X U x U P P x = {P ∈ P : x ∈ P } P P x F = P\P x x /∈ ∪F U P U ∩ ∪F = ∅ x ∈ ∪F\∪F ∪F X k K X K ∩ ∪F K P F ⊂ P {K ∩ P : P ∈ F} = {F 1 , F 2 , . . . , F k } K ∩ ∪F = F 1 ∪ F 2 ∪ . . . ∪ F k i = 1, 2, . . . , n F i = K ∩ P i P i P P i X F i X K ∩ ∪F X K ∩ ∪F K P  X k P X P P P ⇒ P P CF P CF P CF k X P P  P X P P CP ⇒ ⇒ P CP P x ∈ X U x P P x = {P ∈ P : x ∈ P } P P x F = P\P x x /∈ ∪F U P U ∩ ∪F = ∅ x ∈ ∪F\∪F ∪F P CP F ⊂ P  P ∈F P =  P ∈F P ∪F P  P X P P P k X P P P X P P P P P P P ⇒ ⇒ ⇒ ⇔ ⇔ k ⇒ ⇒ k ⇔ k  P X CP WHCP HCP HCF CF P P X P k P ⇒ ⇒ P K X U K P  = {P ∈ P : P ⊂ U} P x ∈ K P ∈ P x ∈ P ⊂ U P ∈ P  x K K ⊂ ∪P  P  K P  K P  K P 0 ∈ P  {P 0 } K x 1 ∈ K\P 0 P  K P 1 ∈ P  x 1 ∈ P 1 {P 0 , P 1 } K x 2 ∈ K\(P 0 ∪ P 1 ) {x n : n ∈ N} x n ∈ K\  n−1 i=1 P i x n ∈ P n x n = x m n = m x n ∈ P n P WHCP {x n : k ∈ N} K {x n } K {x n : n ∈ N} {x n } F ⊂ P  K ⊂ ∪F F ⊂ P  ⊂ P K ⊂ ∪F ⊂ U F P P k X  3. CS ¨o cs ¨o ¨o σ X ¨o σ P =  {P n : n ∈ N} P n ⊂ P n+1 P n n n ∈ N A n = {x ∈ X : P n x} {P \A n : P ∈ P n } A n X {P \A n : P ∈ P n } {P α : α ∈ Λ} ⊂ P n P \A n β ∈ Λ x β ∈ P β \A n x β /∈ A n P n x β {P α : α ∈ Λ} P γ ∈ {P α : α ∈ Λ}\{P β } x γ ∈ P γ \A n P η ∈ {P α : α ∈ Λ}\{P γ , P β } x η ∈ P η \A n {x α : α ∈ Λ} x α ∈ P α \A n P α \A n P n {x α : α ∈ Λ} X ¨o {x α : α ∈ Λ} I ⊂ Λ x /∈ A n x α = x α ∈ I P n x x /∈ A n {P \A n : P ∈ P n } A n A n {z h ∈ A n : h ∈ Γ} A n P n z h h ∈ Γ {P h : h ∈ Γ} P n P h z h ∈ P h h ∈ Γ} P n W HCP {z h ∈ A n : h ∈ Γ} A n X ¨o A n P n  = {P \A n : P ∈ P n } ∪ {{x} : x ∈ A n } P  =  {P n  : n ∈ N} P n  P  P  X x ∈ X U x n ∈ N x ∈ A n x ∈ {x} ⊂ U n x /∈ A n P k ∈ N P ∈ P k x ∈ P ⊂ U x /∈ A n P \A n ∈ P k  x ∈ P\A n ⊂ U P  P  = {F 1 , F 2 , . . . , F n , . . .} n ∈ N y n ∈ F n D = {y n : n ∈ N} D x ∈ X V x k ∈ N x ∈ F k ⊂ V V ∩ D = {y k } x ∈ D x D ⊂ X D = X D X  f : X → Y ¨o X σ f s f : X → Y ¨o P σ W HCP X f ¨o y ∈ Y f −1 (y) ¨o X P  = {P ∩ f −1 (y) : P ∈ P} P  σ W HCP ¨o f −1 (y) f −1 (y) X f s  X P P X k P k P P P ⇒ P P ∈ P P P P P ⇒ P X P P X F ⊂ X F ∩ P P P ∈ P F X X k K X F ∩ K X P k F ⊂ P K ⊂ ∪F ⊂ X F ∩ K = ∪{(F ∩ P ) ∩ K : P ∈ F} F ⊂ P F ∩ P P P ∈ F K P ∈ P P X T 2 K P X (F ∩ P) ∩ K X ∪{(F ∩ P ) ∩ K : P ∈ F} X F ∩ K X X P P ⇒ P K K P x ∈ K P U x x U x P {U x : x ∈ K} K {U x 1 , U x 2 , . . . , U x n } {U x : x ∈ K} K ⊂  n i=1 U x i U x i P i = 1, 2, . . . , n K P P  ¨o X ¨o P X x ∈ X U x x U x P {U x : x ∈ X} ¨o X {U x 1 , U x 2 , . . . , U x n , . . .} {U x : x ∈ X} X =  ∞ n=1 U x n U x n P n = 1, 2, . . . X P P X P  k k X k k f : X → Y Y k f ¨o f cs f ¨o P k k X f(P) k Y f(P) Y K Y U K f L X f(L) = K f f −1 (U) L X P k X F ⊂ P L ⊂ ∪F ⊂ f −1 (U) K ⊂ ∪f (F) ⊂ U f (F) f(P) f(P) k Y f(P) Y K Y K ¨o Y f ¨o f −1 (K) ¨o X P k k X P P  = {P ∩ f −1 (K) : P ∈ P} ¨o f −1 (K) P  = {P ∩ f −1 (K) : P ∈ P} {f(P) : f(P ) ∩ K = ∅ : P ∈ P} K f(P) f(P) k Y f cs P k k X f(P) k Y K Y f(P) P k k X P k X k X Y T 2 K K f f −1 (K) X f −1 (K) X f cs f −1 (K) ¨o f −1 (K) ¨o P P  = {P ∩ f −1 (K) : P ∈ P} ¨o f −1 (K) {P ∩ f −1 (K) : P ∈ P} {f(P) : f(P) ∩ K = ∅ : P ∈ P} K Y f(P)  cs ¨o k k k k k k X k k f : X → Y cs ¨o k k Y k X k k X k k X f cs ¨o Y k Y k k  23 1 35 76 18 99 σ k , 35 (2A) (2005), 5 - 15. CF CS . CF CS (1) CF CF,HCF HCP, CP CF W HCP k cs cs ¨o k k k 1. T 1 X P X P x ∈ X U x P ∈ P x ∈ P ⊂ U P k K U K X F ⊂ P K ⊂ ∪F ⊂ U X P X 1 2 P K X K P P cs A A P P CF K X {K ∩ P : P. Λ x α ∈ P α , α ∈ I  {x α : α ∈ I} =  {x α : α ∈ I}. P x ∈ X U x x U x P P x ∈ X ⇒ cs ⇒ HCF ⇒ CF ⇒ CF ⇒ HCP ⇒ CP WHCP X P X X P U ⊂ X X U ∩ P P P ∈ P X A ⊂ X A X A A X X k X f : X → Y f f(A). 1, 2, . . . , n F i = K ∩ P i P i P P i X F i X K ∩ ∪F X K ∩ ∪F K P  X k P X P P P ⇒ P P CF P CF P CF k X P P  P X P P CP ⇒ ⇒ P CP P x ∈ X U x P P x = {P ∈ P : x ∈ P } P P x F = PP x x /∈

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