[分析] totally bounded定義
我想證：
if A is a subset of M
then A is totally bounded in M (1)
<=> A is totally bounded (2)
n
(1)的定義是：∀ε>0, ∃x_1,...,x_n€M, s.t. A⊆∪ D_M(x_i;ε)
i=1
where D_M(x_i;ε)={z€M：d(z,x_i)<ε}
n
(2)的定義是：∀ε>0, ∃y_1,...,y_n€A, s.t. A=∪ D(y_i;ε)
i=1
where D(y_i;ε)={z€A：d(z,y_i)<ε}
(2)=>(1)是顯然的
可是(1)=>(2)證很久證不出來，因為我無法處理如果選到的是M的點而不是A的點
如何把他代換成A的點
謝謝幫忙！
(以下是一些猜測，不看沒關係)
-----------------------------
我們有：K compact <=> K totally bounded and complete
而如果K是M的子集
我們有 K compact <=> K compact in M →(定義中的open set是open in M)
↓
(定義中的open set是open in K)
而complete本身的定義只跟自己有關
所以我猜：K compact in M <=> K totally bounded in M and complete
因此才又猜 K totally bounded in M <=> K totally bounded
而且我查totally bounded的定義(Marsden的高微, wiki, wiki proof...)
不是(1)就是(2)，所以我才覺得等價
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