※ 引述《PttFund (批踢踢基金只进不出)》之铭言： : ※ 引述《PttFund (批踢踢基金只进不出)》之铭言： : : Show that Lindelof covering theorem is valid in any separable : : metric sapce. : 这边有两个重点, 一个是: Lindelof covering theorem, 这是说: : a set for which every open covering contains a countable : subcovering. 这跟 compact set 的定义有相似之处 (finite subcovering). : 另一个是: separable, 这是说: a set having a countable dense subset. : 例如实数就是一个 separable set (有理数在实数上 dense). Given a seperable metric space (M,d) ,and a set D = {x_1,.....} which is dense in M. Let G = {A_1,....},where each A is B(x_i,r),x_i in D,r in Q. Lemma. If x in S contained in M , where S is open, then at least one ball in G which contains x is contained in S. _ pf. Since S is open ,there exists B(x;r) contained in S. Since x is in D Choose y in B(x;r/2) ∩ D ,choose r/3 < k < r/2 be rational,then B(y;k) is contained in S and is in G. Lindelof covering theorem. pf. Given a set D in (M,d) and S = ∪A be an open covering of D. If x is in D, then there exists A contained in S s.t. x is in A. By lemma,there is at least one ball in G satisfies " x in A_m contained in A " , we choose the one with the smallest index,m = m(x) then ∪ A_m(x) is a countable open covering of D. x in D note:有编号的A是G里的球,没编号的A是别的东西. --

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