Given two sets, one is closed and contains the other, can I find another closed set that contains one of the sets and is included in the other?

Question

I have two sets $A$ and $B$ such that $A\subset B$ and $A$ is closed. Can I find a closed set $C$ such that $A\subset C\subseteq B$? We can discuss the following two cases separately:

• $B$ is closed
• $B$ is open
• $B$ is arbitrary

Thank you for your help!

Answer 1

Consider following topology on $\mathbb R$,

{$\emptyset, \mathbb R, [0,1], [0,1]^c$}

If we take $A=[0,1]$ and $B=[0,1] \cup \{5\}$

Then we cannot find closed set $C$ satisfying your condition.