[微積] Interior, boundary of a set

看板Math作者 (steve)時間14年前 (2011/05/14 17:45), 編輯推噓0(0推 0噓 34→)
留言34則, 6人參與, 最新討論串1/1
Let S= {x1,x2 ... xn} be a nonempty, finite set of points. (A)what is the interior of S? (B)Waht is the boundary of S? (C) Is S open , closed or neither? 答案A是 空集合 (B) S本身 (C) Closed set 這邊我有點不懂 是因為S是離散的點組成的所以interior是空集合嗎？ 要怎麼用定義去理解 另外我想問問這塊地方跟後面的微積分有什麼關係呢 一直不太懂 -- ※ 發信站: 批踢踢實業坊(ptt.cc) ◆ From: 220.132.83.187
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05/14 17:54, , 1F
你用的課本是?
05/14 17:54, 1F
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05/14 18:16, , 2F
不用想太多，就直接用定義去做就好了…
05/14 18:16, 2F
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05/14 18:25, , 3F
我用的課本是Salas 可是定義我不知道怎麼搞
05/14 18:25, 3F
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05/14 18:25, , 4F
之前去翻C&J還有幾本高微也有在講集合的部分
05/14 18:25, 4F
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05/14 18:25, , 5F
搞不太懂這部份與後面微積分的關係
05/14 18:25, 5F
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05/14 18:50, , 6F
引入點集拓樸來敘述δ-ε語言
05/14 18:50, 6F
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05/14 20:13, , 7F
if x is an interior of S, any neighboorhood of x
05/14 20:13, 7F
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05/14 20:14, , 8F
will contain infinte point of S, but S is finte
05/14 20:14, 8F
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05/14 20:15, , 9F
so the interior of S is empty
05/14 20:15, 9F
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05/14 20:15, , 10F
any small enough neighboorhood of x 才對
05/14 20:15, 10F
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05/14 20:16, , 11F
不對 我在搞笑不好意思 這應該是錯的
05/14 20:16, 11F
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05/14 20:20, , 12F
if x is an interior of S, x must be xn for some n
05/14 20:20, 12F
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05/14 20:22, , 13F
since any neighboorhood of x isn't contained in S
05/14 20:22, 13F
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05/14 20:24, , 14F
hence there are no neighboorhood of x such that
05/14 20:24, 14F
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05/14 20:25, , 15F
the neighboorhood of x is contained in S
05/14 20:25, 15F
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05/14 20:25, , 16F
the interior is empty
05/14 20:25, 16F
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05/14 20:28, , 17F
the boundary of S is S unioning the limit point
05/14 20:28, 17F
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05/14 20:29, , 18F
of S, and the limit point of S is empty
05/14 20:29, 18F
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05/14 20:30, , 19F
if x is an limitpoint of S,any neighboorhood of x
05/14 20:30, 19F
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05/14 20:31, , 20F
will contain infinte point of S, but S is finte
05/14 20:31, 20F
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05/14 20:31, , 21F
so the limit point set of S is empty
05/14 20:31, 21F
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05/14 20:31, , 22F
hence the boundary of S is S
05/14 20:31, 22F
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05/14 20:32, , 23F
上面又不對了 又搞笑了
05/14 20:32, 23F
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05/14 22:45, , 24F
拜託大大根我解釋了 還是我該去番什麼書有詳細講
05/14 22:45, 24F
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05/14 22:46, , 25F
你先把定義寫出來再去考慮如何套
05/14 22:46, 25F
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05/14 23:38, , 26F
A point Xo is said to be an interior of the
05/14 23:38, 26F
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05/14 23:39, , 27F
set S if Scontains some neighborhood of Xo
05/14 23:39, 27F
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05/14 23:39, , 28F
要怎麼用
05/14 23:39, 28F
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05/14 23:46, , 29F
所以各點[x1,x2,x3...]之間都有距離
05/14 23:46, 29F
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05/14 23:46, , 30F
取這些可能的距離裡面最小的距離為r
05/14 23:46, 30F
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05/14 23:48, , 31F
則對任何一點來說，它的距離r以內的鄰域(neighboor)
05/14 23:48, 31F
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05/14 23:50, , 32F
都不包含任何一個S集合內的點
05/14 23:50, 32F
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05/14 23:52, , 33F
反過來說就是任何一個xn的鄰域都沒被S包住
05/14 23:52, 33F
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05/14 23:53, , 34F
所以每一個xn都不是S的interior，所以S沒有interior
05/14 23:53, 34F
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文章代碼(AID): #1Dpayv1x (Math)