作者Sfly (entangle)
看板Math
标题Re: [分析] 高微(9)
时间Sat Jul 23 15:51:18 2005
※ 引述《PttFund (批踢踢基金只进不出)》之铭言：
: 请证明存在一函数 f 使得 f: (0,1)╳(0,1) → (0,1) 为 1-1 且 onto.
其实就是在证R跟R^2有一样的势
a=0.a1a2a3....
b=0.b1b2b3....
(舍弃从某位开始都是九的表示法)
f(a,b)=0.a1b1a2b2xxxxxxxxx 即为所求
then f is 1-1, but not onto since 0.919191919191xx isn't in the image
however, R-Q lies in the image of f,
and that im f contains infinitely many rational numbers.
Therefore, there exists g:(0,1) -> (0,1) such that
g leaves all irr fixed, and g maps Q in im(f) to Q 1-1 and onto.
then g(f) satisfies the condition.
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※ 发信站: 批踢踢实业坊(ptt.cc)
◆ From: 220.135.132.108
1F：推 PttFund:Yes140.112.218.142 07/23
2F：推 lianger :觉得有一些无理数不会落在Im(f) 09/25 00:45
3F：→ lianger :比如说0.9999999......与某个无理数交错之後的数。 09/25 00:46