This question was asked in my analysis quiz and I marked every option wrong. I admit I didn't had much clue about this question.
Let $(X,d)$ be a metric space. Then:
A An arbitrary open set $G$ in $X$ is a countable union of closed sets.
B An arbitrary open set $G$ in $X$ cannot be countable union of closed sets if X is connected.
C An arbitrary open set $G$ in $X$ is a countable union of closed sets only if $X$ is countable.
D An arbitrary open set $G$ in $X$ is a countable union of closed sets only if $X$ is locally compact.
Answer
A
I marked B and was clueless about D . But now even if I have answer key I am unable to reason open sets with union of Closed sets and also with locally compact and connected sets .
Can you please give hints for A,B,D.
I shall be really thankful
Best Answer
If we prove that (A) is true it follows immediately that the other options are all false.
Any closed set $C $ in a metric space is the intersection of the sets $\{x: d(x,C) <\frac 1 n\}$ and these sets are open. Taking complements we see that (A) is always true.