作者r4553280 (Q睿)
看板Math
標題[分析] Cardinality、Compact
時間Mon Oct 3 21:58:53 2011
1. Let A be the set of all functions from (0,1) to |R,and
B be the set of all continuous funtions from (0,1) to |R.
(a) Show that there is no 1-1 correspondence between |R and A.
That is,|R and A do not have the same cardinlity.
(b)Prove or disprove that there is a 1-1 correspondence between |R and B.
2.Let W be a compact subset of |R^n and {Va} be an open cover of W.
Prove that there is an ε>0 such that for each subset E of W having
diameter less than ε, there si a V in {Va} containing E.
a0
1.(a)在拓樸課本的習題有看過 說＃（A）＝ 2^（c） , c是|R的個數.
可是仍不知道怎麼證明，其他題則是無頭緒，希望版上得高手能給的指教
小弟感謝萬分！
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→ recorriendo :1(a)如果只要證題目說的用反證就好 假設有一一對應 10/04 05:01
→ recorriendo :那麼可以構造出新函數blabla... 如果要證你說的就比 10/04 05:01
→ recorriendo :較費工 (b)提示:給定連續函數在有理點上的值則此連續 10/04 05:02
→ recorriendo :連續函數已被唯一決定 10/04 05:03
→ recorriendo :2.從定義以及有限開集的交集是開集 不難吧 10/04 05:06
→ jacky7987 :我當初也有寫到第二題 我對題目的認知是說 他會在那 10/04 10:36
→ jacky7987 :堆covering裡面的其中一個 因為我當初想說用cpt 10/04 10:37
→ jacky7987 :找出有限個 但是選出來的那幾個可能沒有那個V_0 10/04 10:37
→ jacky7987 :也就是那個集合是否有可能落在取出來的finite cover 10/04 10:38
→ jacky7987 :中兩個的聯集 10/04 10:38
→ jacky7987 :(就是個交集一些，這種情況是不是又得縮小半徑? 10/04 10:39