[分析] 問一有關連續與反函數的問題

看板Math作者 (哲平)時間13年前 (2011/05/02 11:49), 編輯推噓0(0推 0噓 9→)
留言9則, 2人參與, 最新討論串1/1
我看見書上有一段敘述是如此的 If f: X → Y is continuous and bijective, and its inverse map g: Y → X is also continuous, then f is called a homeomorphism and X and Y are said to be homeomorphic. 我好奇的是它的條件, 我能否找出例子, 說明函數f是bijective跟continuous 但是它的反函數卻是discotinuous的嗎? 小弟高微沒學好, 有請高手給個例子XD -- ※ 發信站: 批踢踢實業坊(ptt.cc) ◆ From: 123.240.168.28
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05/02 12:02, , 1F
let X=Y(as sets), f=id, 並且X的拓樸比Y的細緻
05/02 12:02, 1F
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05/02 14:49, , 2F
上面即例子,熟悉的實數空間也有許多,比如
05/02 14:49, 2F
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05/02 14:50, , 3F
S={0,1,1/2,..},N={0,1,2,.},f:N->S,f(n)=1/n,f(0)=0
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05/02 14:55, , 4F
想要舉|R^n裡的例子的話,如f:S->|R^n (S:|R^n的子集)
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05/02 14:56, , 5F
要注意"至少"不能取 |R^n 的 open set
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05/02 14:57, , 6F
有一個很強的定理 , U:open set in |R^n , f:U－>|R^
05/02 14:57, 6F
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05/02 14:57, , 7F
if f is continous and 1-1 . Then f∣U:U－>f(U)
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05/02 14:57, , 8F
(the restriction of f to U) is a homeomorphism .
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05/02 15:17, , 9F
上面定理中f是 f:U－>|R^n 空格沒看好抱歉
05/02 15:17, 9F
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