问道拓扑题。。XDF模拟#56

weysyUID: 2576906

For a subset S of a topological space X, let cl(S) denote the closure of S in X, and let S'={x: x 属于 cl(S-{x})} denote the derived set of S. If A and B are subsets of X, which of the following statements are true?
I. (A or B)' = A' or B'
II. (A and B)' = A' and B'
III. If A' is empty, then A is closed in X.
IV. If A' is open in X, then A' is not empty.

(A) I and II only
(B) I and III only
(C) II and IV only
(D) I, II, and III only
(E) I, II, III, and IV

答案是B。
I 可以理解是对的，例如A = (0,2) B = (1,3)
II 可以理解是错的，例如A = (0,1) B = (1,3)
III 为什么是对的？
IV 为什么是错的？

拜谢！

zxx117UID: 2681428

A' is empty -> 无limit point, 而一个集合closed等价于包含所有limit point，所以iii正确。

iv的逆否命题：If A' empty then A' is not open in X, 错因为empty set必须为open set。

wlshotUID: 2378155

Very straightforward
'.'  A U A'=cl(A)

.'.  A=cl(A)

.'. A is closed