[分析] 問一有關連續與反函數的問題

[anikishawn (哲平)]

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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→ Sfly        :let X=Y(as sets), f=id, 並且X的拓樸比Y的細緻
→ keroro321   :上面即例子,熟悉的實數空間也有許多,比如
→ keroro321   :S={0,1,1/2,..},N={0,1,2,.},f:N->S,f(n)=1/n,f(0)=0
→ keroro321   :想要舉|R^n裡的例子的話,如f:S->|R^n (S:|R^n的子集)
→ keroro321   :要注意"至少"不能取 |R^n 的 open set
→ keroro321   :有一個很強的定理 , U:open set in |R^n , f:U－>|R^
→ keroro321   :if f is continous and 1-1 . Then f∣U:U－>f(U)
→ keroro321   :(the restriction of f to U) is a homeomorphism .
→ keroro321   :上面定理中f是 f:U－>|R^n    空格沒看好抱歉