想請問 一個證明
有地方不會 有的不太清楚 對不對
抱歉 麻煩大家了
Let x_1,x_2,.. be a sequence of the points of the product space
ΠX_a. Show that this sequence converges to the point x
<=>
the sequence Π_a(x_1),Π_a(x_2),.. converges to Π_a(x) for each a.
Pf：
“=>”
If U is open and Π_a(x) belong to U
then (Π_a)^(-1)(U) is open and x belong to (Π_a)^(-1)(U)
for some N≧1
s.t x_n belong to (Π_a)^(-1)(U) where n≧N
For such n , Π_a(x_n) belong to U.
請問 這部份這樣寫 對嗎？
“<=”
Let V 包含於 ΠX_a be open set contains to x
W=ΠW_a, where W_a=X_a for some a_i,i=1,2,. ,k
s.t W_a_i is open in X_a_i
and W_a=X_a for other indices
s.t x belong to W包含於V
X_a_1 belong to W_a_1, … , X_a_k belong to W_a_k
這地方接下來不知道該如何證
想請指點指點謝謝^.^
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