※ 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.
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※ 編輯: hips 來自: 218.174.181.77 (08/21 22:49)