[分析] compact一問

[mathQ (math_J)]

以下幾個範圍 要證明K belong R 是否 compact

(a) K=[0,1]

(b) K=(0,1)

(c) K={1/n : n belong N} U {0}

然後題目要求 using the definition of compactness



我一開始直覺只想到要用closed bounded <=> compact

c那題也只單純運用 1/n的極限點=0 去證

但仔細看題目上面說

" using the definition of compactness "

就頓時猶豫起來要如何用compact本身的定義去pf



另外問一題

b(A) 交集 b(B) = b(A交集B)

此處b(A) 表 the boundary of a set A belong R^n




麻煩各位了  m(_ _)m

謝謝

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◆ From: 59.115.4.225
→ Sfly        :c不是compact. 你用 1/n的極限點=0證了什麼?
→ Sfly        :sorry 別理我
→ kane950544  :bdd&closed iff compact,a&c are compact
→ kane950544  :b is not.counterexample: Union (0,1-(1/n))
→ kane950544  :[n=1,2,...] covers K, but doesn't have finite
→ kane950544  :subcovering
推 znmkhxrw    :另外問得那題是錯的吧? A取有理數集合 B取無理數集合
推 ss1132      :c是compact喔
推 avesta      :a的證明Rudin裡面關於cpt的章節有
推 zombiea     :c) for a covering, find one which 0 lies in it
→ zombiea     :then there's only finite many 1/n not in that set
→ THEJOY      :c)用阿基米德就可以把某個n以後的1/n用一顆球處理掉
→ THEJOY      :剩下n-1個最多只要n-顆球1就能蓋住，所以只要有限個