Looking for two closed sets satisfying some properties

Question

Given $\mathbb{R}^n$ and the standard topology, I wonder if there exist two closed sets $A$ and $B$ satisfying the conditions:

• $A\cup B=\mathbb{R}^n$
• $C\subseteq A\subseteq D$
• $B\cap C=\emptyset$

where $C$ is a given closed set and $D$ is also given (not necessarily closed).

Answer 1

If $C \cap \partial D = \emptyset$, then we can take a closed $A$ such that $C \subseteq A \subseteq D$ and $\partial C \cap \partial A = \emptyset$. To see this, start by observing that $C$ is contained in the interior of $D$, and since $C$ is closed, we can take an open $O \subset D$ such that $d(O,\partial D) > 0$. Then take $A = cl(O)$. Then take $B= cl(\mathbb{R}^n\setminus A)$.

However, if $C \cap \partial D \neq \emptyset$, this is impossible. Taking $x$ in that intersection, there would have to be a sequence of points in $B$ that approached $x$. Since $B$ needs to be closed, we would have $x \in B \cap C$.

Thus, such $B$ and $C$ exists if and only if $C \cap \partial D = \emptyset$

Answer 2

In general, no : if $C = D$, then $A = C$. Therefore we need to find $B$ a closed set such that $ A \cup B = \mathbb{R}^n$ and $A \cap B = \emptyset$. That's impossible since $\mathbb{R}^n$ is connected.