UC知道帮忙解下这个数分的题
来源：百度知道 编辑：UC知道 时间：2024/06/18 03:56:42
Given are the following subsets of the Euclidean space R2 :
S1={(x ,0): 0<x<= 1} s2={(1/n ,y): n=1,2,3,... , 0<=Y<= 1}and S3={(0,1/ 2)} Consider the sets S = S1 ∪ S2 , S 0= S1 ∪ S2 ∪ S3
(a) Use the Bolzano-Weierstrass Theorem to prove that the set S0 is not compact.
(b) Is the set S arcwise connected? Justify your answer
(a) Is the set S0 connected? Justify your answer.
S1={(x ,0): 0<x<= 1} s2={(1/n ,y): n=1,2,3,... , 0<=Y<= 1}and S3={(0,1/ 2)} Consider the sets S = S1 ∪ S2 , S 0= S1 ∪ S2 ∪ S3
(a) Use the Bolzano-Weierstrass Theorem to prove that the set S0 is not compact.
(b) Is the set S arcwise connected? Justify your answer
(a) Is the set S0 connected? Justify your answer.
(a)直接构造一个收敛到X轴以外的横坐标是无理数的序列即可。
(b)是。直接构造U型路径即可。
(c)不是。(0,1/2)和其它点不连通，可以用反证法证明，同样是在路径上找一个无理点。