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Set \(\rightarrow\) Space

(metrix\(\leftrightarrow\)map)

Exa:

1.\(\rho(x,y)=x+y\)?

(x)三角不等式：$$\rho(x, z) \leq \rho(x, y)+\rho(y,z)$$

2.\(\rho(x,y)=x \times y\)?

(x)三角不等式：$$\rho(x,z)\leq \rho(x, y)+\rho(y,z)$$

3.一个很有意思的天生的三角关系:距离

\(\sqrt{x^2+y^2}\)

\[\rho\left(\left(x_{1}, x_{2}\right),\left(y_{1}, y_{2}\right)\right)=\max \left\{\left|x_{1}-y_{1}\right|,\left|x_{2}-y_{2}\right|\right\} \]

利用绝对值不等式，把绝对值压在了最下面，这个构造还挺有意思的，这个其实不容易想出来，\(x=(x_1,x_2)\)

5.$$\rho(x, y)=\left{\begin{array}{ll}
0 & \text { if } x=y \
1 & \text { if } x \neq y
\end{array}\right.$$

这个主要是在\(x=y\)，将这个弄到0.....

\[\rho(x, y)=\left\{\begin{array}{ll} 1 & \text { if } x=y \\ 0 & \text { if } x \neq y \end{array}\right.\]

下面这个就不成立了

\[B\left(x_{0}, \epsilon\right)=\left\{x \in X \mid \rho\left(x_{0}, x\right)<\epsilon\right\} \]