请问一道拓扑题

zephyr007UID: 2108380

Let R be the set of real numbers with the topology generated by the basis
{[a,b): a<b,where a,b∈R}    If X is the subset [0,1] of R ,which of the following
must be true?
I    X is compact
II   X is Hausdorff
III  X is connected
(A)  I only
(B)  II only
(C)  III only
(D)  I and II
(E)  II and III

显然 II 正确   III不对
请问为什么 I不对？

jxegongUID: 199307

任一开覆盖存在有限子覆盖是紧，[0,1-1/n)(n是正整数)和[1,2)是[0,1]的开覆盖，但不存在有限子覆盖，所以不是紧的