[分析] compact一問

看板Math作者 (math_J)時間14年前 (2012/02/13 13:54), 編輯推噓4(4推 0噓 9→)
留言13則, 7人參與, 最新討論串1/1
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 -- ※ 發信站: 批踢踢實業坊(ptt.cc) ◆ From: 59.115.4.225
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02/13 14:00, , 1F
c不是compact. 你用 1/n的極限點=0證了什麼?
02/13 14:00, 1F
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02/13 14:01, , 2F
sorry 別理我
02/13 14:01, 2F
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02/13 14:20, , 3F
bdd&closed iff compact,a&c are compact
02/13 14:20, 3F
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02/13 14:23, , 4F
b is not.counterexample: Union (0,1-(1/n))
02/13 14:23, 4F
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02/13 14:24, , 5F
[n=1,2,...] covers K, but doesn't have finite
02/13 14:24, 5F
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02/13 14:24, , 6F
subcovering
02/13 14:24, 6F
推
02/13 14:57, , 7F
另外問得那題是錯的吧? A取有理數集合 B取無理數集合
02/13 14:57, 7F
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02/13 16:00, , 8F
c是compact喔
02/13 16:00, 8F
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02/13 16:15, , 9F
a的證明Rudin裡面關於cpt的章節有
02/13 16:15, 9F
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02/13 17:58, , 10F
c) for a covering, find one which 0 lies in it
02/13 17:58, 10F
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02/13 17:59, , 11F
then there's only finite many 1/n not in that set
02/13 17:59, 11F
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02/13 23:53, , 12F
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
02/13 23:53, 12F
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02/13 23:54, , 13F
剩下n-1個最多只要n-顆球1就能蓋住，所以只要有限個
02/13 23:54, 13F
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