Copestone 發問時間: 科學數學 · 1 0 年前

〔拓璞〕Noetherian 空間之刻劃

稱一個拓璞空間 X 為 Noetherian 的意思是說:

對任意 X 之 descending chain of closed subsets {F1, ..., Fn, ...}〔即後者為前者的子集〕,存在某 N,使得當 i > 或 = N 時,Fi = Fi+1.

證明 X 為 Noetherian if and only if 任何 X 的 open subsets 都為 compact。

請舉出一個 non-trivial 〔即非 finite and discrete〕的 Noetherian 空間。

註:本來想打 「Fi+1 為 Fi 的子集」內的子集符號,只是奇摩的符號老是變亂碼,怕怕。無人回答將自行刪除,對認真回答者我將認真評價或評論。

3 個解答

評分
  • 1 0 年前
    最佳解答

    1. Suppose every open set in X is compact, and {G_k} is a sequence of ascending open sets.

    Let G= ⋃ G_k. Then G is open and {G_k} is an open covering of G.

    By compactness of G, G_n = G for some n.

    Therefore G_k=G_n=G for all k>=n.

    Thus X is Neotherian.

    2. Suppose X is Neotherian, G is an open in X, and {H_k} is an open covering of G.

    Let G_k = ⋃ {H_1, H_2, ... H_k}.

    Then {G_k} is an open covering of G, and it is a sequence of ascending open sets. Since X is Neotherian, there exist an integer n such that G_k=G_n for all k>=n, hence G_n=G, and {H_1, ... H_n} is a finite subcovering. Therefore G is compact.

    2007-05-17 00:50:34 補充:

    修正一下 2. 因為 open covering 不一定 countable, 所以應該用反證法比較好:

    2007-05-17 00:51:58 補充:

    Suppose X is Neotherian, G is an open but non-compact subset in X. There exits and open covering {H_α} of G that has no finite

    subcovering.

    2007-05-17 00:52:54 補充:

    We *can* choose a sequence {H_k} such that, by letting G_k = ⋃ {H_1, H_2, ... H_k, G_k is an *strictly* ascending sequence of open sets. Contradiction to X being Neotherian.

    2007-05-17 11:45:07 補充:

    補充不能一次寫太多。你不是說大家都是有素養的人,重點寫一寫就好了嗎?

    2007-05-17 11:52:04 補充:

    Pick H_{k+1} out of {H_α} in such a way that

    (i) H_{k+1} is not a proper subset of ⋃ {G_k}

    (ii) G_{k+1} = H_{k+1} ⋃ G_k is not an open covering of G.

    This is possible bacause {H_α} has no finite subcovering of G.

  • Eric
    Lv 6
    1 0 年前

    Noetherian 空間的例子:

    On any set X, we can define the topology

    T = {A 包含於 X: X\A is finite or A is empty},

    which turns X into a Noetherian space.

  • 1 0 年前

    你「only if」部份要把它寫完麼?can select ...那也太不嚴謹了吧。

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