Anonymous
Anonymous asked in 科學數學 · 9 years ago

高微有關interior closure的問題

請問有關這兩小題應該要怎麼證明或反證他?

圖片參考:http://imageshack.us/m/804/6972/34171504.jpg

Update:

謝謝解答

不過我可以問一下你這兩小題是怎麼想的嗎?

你第一小題怎麼想的到這個反例?

你是由條件觀察到什麼才舉得出這個反例的?

那第二小題你證明的思路歷程大概是怎樣的?

你怎麼會想到這樣證?

我想瞭解你怎麼證的這樣遇到類似題才會....

Update 2:

那我加一個條件A is a connected set 這樣A的敘述會變正確嗎?那要怎麼證?還是也有其他反例?

1 Answer

Rating
  • Sam
    Lv 6
    9 years ago
    Favorite Answer

    圖片參考:http://imgcld.yimg.com/8/n/AC06918685/o/1511051709...

    (a) F.Let A=(0,1) U (1,2).Then A_bar=[0,2],int(A_Bar)=(0,2) is not equal to A. (b) T.If x in A ∩B_bar, then x in A,and there exists a sequence {bn}[n=1 to infinity]in B such that bn->x.SinceA is open , x in A and bn->x , there is a N such that for all n >= N, bnin A.Sothere exists a sequence {bn}[n=N to infinity] in A∩B, and bn -> x.Hencex in (A∩B)_bar.[[Done]]

    2011-05-18 13:46:09 補充:

    怎麼會想到這樣證?

    不過我可以問一下你這兩小題是怎麼想的嗎?你第一小題怎麼想的到這個反例?你是由條件觀察到什麼才舉得出這個反例的?

    [[Answer]]

    從簡單的R開始想,一個開區間,二個分很開的開區間,二個相鄰的開區間,答案就出來了,有點經驗,有點運氣。

    [[Done]]

    那第二小題你證明的思路歷程大概是怎樣的?

    你怎麼會想到這樣證?

    [[ANSWER]]

    一看,覺得應該對。就直接證明,證明前觀察,此題之關鍵點是什麼?發現是bn 是否在A與B之交集?當然到此,答案已呼之欲出,只剩如何將他清楚地寫出來。

    [[done]]

    那我加一個條件A is a connected set 這樣A的敘述會變

    2011-05-18 13:55:26 補充:

    那我加一個條件A is a connected set 這樣A的敘述會變正確嗎?

    [[Answer]]

    F.

    Let A be the open unit circle deleted the center, and C the open unit circle, and C_bar the closed unit circle..

    Int(A_bar)=int( C _bar)=C is not equal to A.

    Maybe it will be true, if A is a simply connected open set.

    You will try to prove it

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