Re: [高微] A,B closed 判斷A+B是否closed
※ 引述《IminXD (Encore LaLa)》之銘言：
: 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
reference：
https://math.stackexchange.com/questions/216334/closed-subsets-a-b-subset-mathbbr2-so-that-ab-is-not-closed
考慮A=Z^+，B={n+1/(n+1)n|n屬於Z^+}
則B-A={1/(n+1)|n屬於Z^+} is not closed.
現在令A'=-A，B'=B both closed
則A'+B'is not closed.
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