Set ( ightarrow) Space
(metrix(leftrightarrow)map)
Exa:
1.( ho(x,y)=x+y)?
(x)三角不等式：$$ ho(x, z) leq ho(x, y)+ ho(y,z)$$
2.( ho(x,y)=x imes y)?
(x)三角不等式：$$ ho(x,z)leq ho(x, y)+ ho(y,z)$$
3.一个很有意思的天生的三角关系:距离
(sqrt{x^2+y^2})
[
holeft(left(x_{1}, x_{2}
ight),left(y_{1}, y_{2}
ight)
ight)=max left{left|x_{1}-y_{1}
ight|,left|x_{2}-y_{2}
ight|
ight}
]
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.$$
ho(x, y)=left{egin{array}{ll}
0 & ext { if } x=y
1 & ext { if } x
eq y
end{array}
ight.$$
这个主要是在(x=y)，将这个弄到0.....
[
ho(x, y)=left{egin{array}{ll}
1 & ext { if } x=y \
0 & ext { if } x
eq y
end{array}
ight.]
下面这个就不成立了
[Bleft(x_{0}, epsilon
ight)=left{x in X mid
holeft(x_{0}, x
ight)<epsilon
ight}
]