→ ert0700 :在測度空間中有B(x,1/n) is open 和阿基米得原理 12/05 21:28
→ ert0700 :可對任意集合裡的點作逼近,但一般拓墣空間作不到 12/05 21:29
→ ert0700 :我個人理解是這樣 12/05 21:30
※ 引述《ppia ((= =))》之銘言：
: 標題: Re: [分析] 拓樸空間裡的閉集
: 時間: Sat Dec 5 12:33:43 2009
:
: ※ 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.
:
: --
: ※ 發信站: 批踢踢實業坊(ptt.cc)
: ◆ 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 12/05 12:51
: → ppia :什麼問題? 12/05 16:28
: → wuxr :{0} 在這裡面是compact 嗎? 12/05 18:06
: 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.
謝謝P大的解釋, 我還有幾個問題是這樣的
就是對一般的拓樸空間, 能不能用極限來刻劃閉集和緊緻集的特徵
我這樣的想法不知道對不對
(X,T) is a topological space. A is a subset of X,
we say x is a limit point of A iff.
V, any NBD of x, intersects A as nonempty.
(1) F is closed iff. if x is a limit point of F, then x is in F.
I think it is true, is it?
(2) K is compact, then
for any subset A of K, there is a point x in K that is a limit pt. of A
我有兩個問題是
證明(2), 我沒有用到跟K有關的任何事. 後來找書發現書上這樣說
K is compact,then
(*) for any infinite subset A of K, there is a point x in K such that
any NBD of x intersect A as a infinite set.
這裡會用到K 存在有限開子覆蓋這件事. 但是我以(*)要推回K 是closed
卻又做不動.
我知道 compact => closed 一般空間中不一定對
但是從極限的角度看到底哪裡不一樣??
先跟您說聲謝謝,求教了
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 61.217.153.121