作者anikishawn (哲平)
看板Math
標題[分析] 問一有關連續與反函數的問題
時間Mon May 2 11:49:06 2011
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
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1F：→ Sfly :let X=Y(as sets), f=id, 並且X的拓樸比Y的細緻 05/02 12:02
2F：→ keroro321 :上面即例子,熟悉的實數空間也有許多,比如 05/02 14:49
3F：→ keroro321 :S={0,1,1/2,..},N={0,1,2,.},f:N->S,f(n)=1/n,f(0)=0 05/02 14:50
4F：→ keroro321 :想要舉|R^n裡的例子的話,如f:S->|R^n (S:|R^n的子集) 05/02 14:55
5F：→ keroro321 :要注意"至少"不能取 |R^n 的 open set 05/02 14:56
6F：→ keroro321 :有一個很強的定理 , U:open set in |R^n , f:U－>|R^ 05/02 14:57
7F：→ keroro321 :if f is continous and 1-1 . Then f∣U:U－>f(U) 05/02 14:57
8F：→ keroro321 :(the restriction of f to U) is a homeomorphism . 05/02 14:57
9F：→ keroro321 :上面定理中f是 f:U－>|R^n 空格沒看好抱歉 05/02 15:17