Proof:(Heine-Borel) 首先呢,我们可以直接根据Lindelöf Covering Theorem来去作为一个引理,来说:一定有一堆可数的,F的子集{I1,I2,...}覆盖集合A。 接下来我们来考虑有限的并集:Sm=⋃k=1mIk(m≥1) 因为每一个小的Ik全部都是开集,所以马上可以知道Sm一定也是一个开集,我们要做的工作其实很简单明了:证明...
Heine-Borel Theorem Closed and bounded in Rn is equivalent to compact. Proof: Assume E is compact, which means that every open cover of it has a finite subcover. Apply this to any arbitrary covering of E by rational open balls, and since a finite subcover exists, the subcover is bounde...