Re: [分析] closed set
※ 引述《Madroach (∞)》之銘言：
: A and B are nonempty closed subset in R.
: Define A+B = {a+b | a in A, b in B}
: A*B = { ab | a in A, b in B}.
: Prove or disprove A+B, A*B are closed.
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◆ From: 111.248.9.101
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ss大大提供的集合沒錯, 可是原因好像有點怪, 應該是這樣:
For all positive integer n ≧ 2, we see that n belong A and
1 1 -1
–n–── belong B, so n ＋ (–n–──) ＝ ── are in A + B for all n ≧ 2.
n n n
1
But ｌｉｍ ── ＝ 0 does not belong A + B. ( A + B has no integers.)
n→∞ n
至於 A ×B = { ab | a in A, b in B.} 是不是 closed 呢? 答案不是. 反例如下:
1 +
Set A ＝ Ｚ, and B ＝ {──｜n is in Ｚ .} ∪ {0}. We have already known
n
that both A and B are closed in |R. But A ×B ＝
q __
{──｜p, q are both in Ｚ, p ≠0.} ＝ Ｑ is not closed in |R due to Ｑ
p
＝ |R.
(題目來源: 台灣聯合大學系統 99學年度碩士班考題 高等微積分 #2)
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: 數學到底有什麼技巧呢?
靈性, 信念, 經驗
: 想不出來做不出來是真的不會嗎?
我覺得 這是緣分的問題 (茶)
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※ 編輯: sato186 來自: 114.33.209.112 (02/10 04:08)
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02/10 08:34, , 1F
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