Re: [微積] 兩題高微

[k6416337 (とある煞氣の光希)]

※ 引述《viviviccc (祖)》之銘言：
: 小弟最近要交功課遇上了兩個問題
: 課本後面也沒有詳解
: 所以想上來MATH版向各位高手請教一下該怎麼解題
: http://i.imgur.com/DXmBj.jpg
: 8.8.8B小題

8.3.8B
"=>"True.
Let V be closed. For all x∈V, define a closed ball B_x=B(x,d(x,bd(V))).
[p.s. d(x,S)=inf{||x-y|| :y∈S}, bd(V)=boundary of V]
We will show that V=∪_(x∈V) B_x.
"ㄈ" Trivial.
"コ" Assume that there is some x∈V such that B_x is not contained in V.
     Then it means y!∈V for some y∈B_x. Since V is closed, y is an exterior
     point of V.
     Consider the line segment L connecting x and y. Then L must intersect V
     by a boundary point, say z. Thus, we have the result that
           d(x,bd(V))≧||x-y||>||x-z||≧d(x,bd(V)), →←
     Therfore, B_xㄈV for all x∈V => Vコ∪_(x∈V) B_x.
By the two proofs, V=∪_(x∈V) B_x.

"<="False.
Counterexample:n=1, B_k=[0,1-1/k] for all intergers k>1.
               Then V=∪_(k>1) B_k=[0,1) is not closed.

: http://i.imgur.com/RU0uo.jpg
: 8.4.11A小題

U is relatively open in E <=> U=O∩E for some open set OㄈR^n.
Since UㄈE^0, U=O∩E^0. [otherwise U contains some boundary points of E]
Then U is open in R^n. Thus, U∩bd(U)=ψ.

: 可以的話麻煩盡量詳細一點
: 小弟只有微薄1000P幣可以報答
: 謝謝了

沒問題的話，1000P可以入戶了

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