jimlucky的部落格

Let X be a compact Hausdorff
C(X,R)={f|f:X->R is bdd and conti}
show the weak topology on X generated by C(X,R)
equals the original topology.


寫了整整一星期，不過幾乎都花在把問題搞清楚
(腦袋太差，都看不懂><)

後來想清楚後動筆只花了一晚就拼出來了~~

雖然上台講時有點瑕疵，不過大致想法是對的~^^

寫出來還挺爽的~

每次想很久，結果寫下來卻只有幾行就KO了@@

proof:

It suffices to show every closed set is weakly closed
(why?)
let E is closed
for every x belong to complement of E
by Urysohn Lemma
there exist a function fx belong to C(X,R) s.t
fx(x)=1,fx(E)={0}

so
x belong to Tx(1/2,3/2)( =inverse image of fx((1/2,3/2)) )
and Tx intersection E is empty
so Tx((1/2,3/2))belong to complement of E
so complement of E = union of Tx((1/2,3/2))
since union of Tx((1/2,3/2)) is weakly open
so E is weakly closed. QED