作者anikishawn (哲平)
看板Math
标题[分析] 问一有关连续与反函数的问题
时间Mon May 2 11:49:06 2011
我看见书上有一段叙述是如此的
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.
我好奇的是它的条件,
我能否找出例子,
说明函数f是bijective跟continuous
但是它的反函数却是discotinuous的吗?
小弟高微没学好,
有请高手给个例子XD
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1F：→ Sfly :let X=Y(as sets), f=id, 并且X的拓朴比Y的细致 05/02 12:02
2F：→ keroro321 :上面即例子,熟悉的实数空间也有许多,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 :想要举|R^n里的例子的话,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 :有一个很强的定理 , 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 空格没看好抱歉 05/02 15:17