平面上的点和直线上的点一样多

$\mathbb{R}^2$和$\mathbb{R}$之间可以形成双射.

由于$\mathbb{R}^2$可以和$[0,1]\times [0,1]$形成双射，而$\mathbb{R}$可以和$[0,1]$形成双射，因此我们只用证明

 

$[0,1]\times [0,1]$可以和$[0,1]$形成双射.

设$A=[0,1],B=[0,1]$.我们要证明$A\times B$和$[0,1]$可以形成双射.由于$[0,1]$可以和$2^{\mathbb{N}}$形成双射,因此我们只用证明

$A\times B$可以和$2^{\mathbb{N}}$之间形成双射.w

首先易知存在从$2^{\mathbb{N}}$到$A\times B$的单射,根据Cantor-Bernstein-Schroeder定理,我们只用证明存在从$A\times B$到$2^{\mathbb{N}}$的单射.我们可以把$2^{\mathbb{N}}$看作所有0-1序列.我们下面来看这个图:

 

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以一种特定的方式构造从$2^{\mathbb{N}}\times 2^{\mathbb{N}}$到$2^{\mathbb{N}}$的单射是很容易的.完毕.