I have another question. I am trying to show that the subset (0,1](0,1] is not compact in RR with the usual topology. Note that ⋃n∈N(1n,n)⋃n∈N(1n,n) covers (0,1](0,1]. Thus, if (0,1](0,1] were compact, then there would exist {n1,...,nk}⊆N{n1,...,nk}⊆N such that ⋃ki=1(1ni,ni)⋃i=1k(1ni,ni). WLOG, take 1n1≤1ni1n1≤1ni for every ii. Hence, there exists an m∈BbbNm∈BbbN such that 1m<1n11m<1n1 and...
therefore an element in (0,1](0,1] but not in not in ⋃ki=1(1ni,ni)⋃i=1k(1ni,ni), a contradiction.
Does this sound right?