Re: [分析] 拓樸空間裡的閉集

[ppia ((= =))]

※ 引述《wuxr (wuxr)》之銘言：
: 請問各位先進一個問題
: If (X, T) is a topological space,
: A is closed iff every conv. seq. in A converges to a point in A.
: 由左到右我有寫出來, 但是由右到左我被卡住了
: 還是說這個命題只有單邊的(左到右)
: 求教了!謝謝^^

有反例:
Put X=|R, T={ (-n,n) | n=0,1,2,3...} u {|R}.

{0} is not closed since |R-{0} = (-∞,0)u(0,∞).
However, the only sequence in {0} is {0,0,...},
which indeed converges to 0.

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◆ From: 59.104.172.114
推 wuxr        :喔~~我了解了...謝謝P大...能不能再請教一個問題
→ ppia        :什麼問題?
→ wuxr        :{0} 在這裡面是compact 嗎?
                是

compact => closed 在非Hausdroff空間中不一定對
(Hausdroff: For any points x and y in X, there exist two disjoint
neighborhoods U and V of x and y, respectively.)

不過上面的例子即使在Hausdroff空間中還是可以找到

Put X=|R, T={ S | S is dense* in (infS, supS).} (* dense in the usual sense.)
(X,T) is a Hausdroff topological space.

Set A=|R-|N, which is not closed since |N is not dense* in |R.
Let {x_n} be an arbitray sequence in |R. Seeing that |R-{x_n} is dense* in |R,
which means |R-{x_n} being open, the sequence cannot converge to a point out
of {x_n}. Therefore, any convergent sequence in A must converge to a
point in A.
※ 編輯: ppia            來自: 59.104.172.114       (12/05 19:29)