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ALSON asked in 科學數學 · 1 decade ago


1. A為閉集⇔A包含其所有聚點

2. cl(A)=A∪accu(A)

3. x∈cl(A) ⇔inf{d(y,x)│y∈A}=0

4. 設{Xn}為Cauchy序列,則 (1){Xn}為有界 (2)若存在子序列Xnk→X⇒Xn→X

≠ < > ≤ ≥  | | ! ~ ⇒

⇔ →  ↔  ∧ ∨ ⊕ ∀ ∃

≡ ∈ ⊆ ⊂ ⊇ ⊃ ∪ ∩ \ ∞

π || || ∑ ∏ ∫ ∇ ∂ ⊥ ≈ √

∵ ∴ ε ω ρ σ α β γ δ

1 Answer

  • 1 decade ago
    Favorite Answer

    1.(=>)Suppose A is closed,then M\A is open(M denoted the metric space),Now if x€M\A,there is ε>0 such that B(x,ε)∩A=空集合

    so x is not an accmulation point of A

    so A contains all its accumulation points

    (<=)Suppose A contains all its accmulation points,if x €M\A,x不屬於A and x is not accmulation point of A

    so there is ε>0,such that B(x,ε)∩A=空集合

    =>B(x,ε) contained in M\A

    =>M\A is open

    =>A is closed

    2Let B=A∪accu(A),Note that for any C is closed set containing A must be contains B

    since if x€B,then x€A or x is an accmulation point of A,x€A=>x€C

    x is an accmulation point of A,for ε>0,the open set B(x,ε)∩A\{x} is not empty

    and B(x,ε)∩C\{x} is also not empty,x is accmulation point of C,C is closed


    Hence if B is closed,then it must the smallest closed set containing A

    and b=cl(A)

    so it suffices to show that B is closed

    Consider y is an accmulation point of B,for ε>0,B(y,ε) contains other point of B,say z,then z€A or z is an accmulation point of A

    In the latter case,choose δ=ε-d(y,z),then B(z,ε-d(y,z)) contains other point of A,say z',we have z'≠y,if not ,then z' can not be belong to B,a contracdits our hypothesis

    so z'≠y,and d(y,z')<=d(y,z)+d(z,z')<ε

    =>y is an accmulation point of A


    2008-01-17 17:33:31 補充:

    第三題的證明 你要先知道以下這個事實:


    3(only if) Let x€cl(A),by definition,for any η>0,choose y€A,such that d(x,y)<η

    this is implies that inf{d(x,y)|y€A}=0

    (if) Suppose that inf{d(x,y)|y€A}=0

    given anyη>0,choose y€A,such that d(x,y)<0+η=η


    Source(s): 剩下的明天再寫
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