黑桃ㄟ死 發問時間: 科學數學 · 1 0 年前

高微(perfect set)

Is there a nonempty perfect set in R which contains no rational number?

2 個解答

評分
  • L
    Lv 7
    1 0 年前
    最佳解答

    Yes, 因為 |R 任意扣掉一個 measure zero 的集合後一定會包含

    一個完美集

    2006-08-30 00:53:41 補充:

    the set of rational number, say Q, is measure zero,

    thus there is a collection of open intervals {I_n} such that

    U I_n convers Q and Σ|I_n| < 1

    then |R - U I_n has positive measure,

    which implies that |R - U I_n is a uncountable closed set.

    hence Cantor-Bendixon theorem say that

    |R - U I_n = the union of a perfect set and a countable set

    so we find a perfect set is contained in |R - Q

    參考資料: ME
  • 1 0 年前

    利用 Cantor 定理是可以很快證明一個不包含任何有理數的 perfect 集合,不過對於實數域而言,不必用到此定理,而可以直接建構出此種集合。

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