※ 引述《PttFund (批踢踢基金只進不出)》之銘言： : ※ 引述《PttFund (批踢踢基金只進不出)》之銘言： : : Show that Lindelof covering theorem is valid in any separable : : metric sapce. : 這邊有兩個重點, 一個是: Lindelof covering theorem, 這是說: : a set for which every open covering contains a countable : subcovering. 這跟 compact set 的定義有相似之處 (finite subcovering). : 另一個是: separable, 這是說: a set having a countable dense subset. : 例如實數就是一個 separable set (有理數在實數上 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:有編號的A是G裡的球,沒編號的A是別的東西. --

※ 發信站: 批踢踢實業坊(ptt.cc) ◆ From: 218.174.181.77 ※ 編輯: hips 來自: 218.174.181.77 (08/21 22:49)