[高微] A,B closed 判斷A+B是否closed

[IminXD (Encore LaLa)]

題目是: Let A,B ≦ R^n be closed set.
        Does A+B = {x+y | x€A and y€B} have to be closed?


解答是找例子 A={(x,0) €R^2 | x€R}
             B={(t,1/t) €R^2 | t>0}



我自己是從定義著手

A is closed => (R^n - A) is open
B is closed => (R^n - B) is open

to defined whether A+B is closed or not,we consider R^n-(A+B)

R^n-(A+B) = R^n-A-B = (R^n-A)-B

since R^n-A is open , and B is close

=> R^n-(A+B) is not open

Hence A+B is not closed ##


問題點就是說 open - closed = closed  能不能推過去..

感覺這個也是要舉例證明.....囧

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◆ From: 114.42.103.126
→ Sfly        :A+B={(x,y)|x>0} is open.
推 wyob        :不可能定義做吧，因為還是有close+close=close的阿
→ wyob        :所以直接舉反例吧
→ IminXD      :我是從題目要我判斷是否closed，但我找到open  這樣
→ IminXD      :S大好像只看標題就推文了XD  已經有定義A+B..
→ Sfly        :我是只看到hint. 你的證明是錯的  這題是要舉反例
→ tiwsjia     :反例：A = -|N, B = {n+1/n: n > 2}
→ IminXD      :樓上的 A = -|N 是什麼意思阿QQ?
→ tiwsjia     :-|N = {-1,-2,-3,...,-n,...}
→ yueayase    :R^n-(A+B)=(R^n-A)-B=(R^n-A)交集(R^n-B)
→ yueayase    :R^n-A 和 R^n-B都open, 有限open set交集是open
→ yueayase    :好像你對??
→ yueayase    :啊不對,R^n-(A+B) = R^n-A-B?? 不是吧??
推 plover      :要證明 "=> not open"，必須要舉反例證明 XD
→ plover      :以前算證明我也會繞來繞去