A question about open subsets of the plane whose components are the interiors of circles. We assume the Axiom of Choice. Let the term "Bubble Set" denote a proper non-empty open subset of the Euclidean plane P, all of whose components are the interiors of circles-i.e. open disks-which are subsets of P. If B is a Bubble Set, does there always exist a Bubble Set B* such that (1) B* is dense in P and (2) B is a subset of B*? It looks almost intuitively obvious that the answer should be "yes". If B is not already dense in P, then we can always enlarge B by adding another open disc- and this process must end somewhere. But I cannot seem to be able to come up with a proof that really satisfies me.

Let $B$ be given. Let $U=P\setminus \overline B$. Let $\{p_i:i\in\omega\}$ be a dense subset of $U$. Let $D_0$ be an open disk with $p_0\in D_0\subseteq U$. Let $i_1=\min\{i\in\omega:p_i\notin \overline D_0\}$. Let $D_1$ be an open disk with $p_{i_1}\in D_1\subseteq U$ and $D_1\cap D_0=\varnothing$. Continue this process, recursively defining a collection of pairwise disjoint open disks in $U$ such that every $p_i$ is in the closure of some $D_n$.  That is, assuming $i_k$ and $D_k$ have been defined for $k<n$, let $i_n$ be the least integer such that $p_{i_n}$ is not in the closure of any $D_k$, and let $D_n$ be an open disk  with $p_{i_n}\in D_n\subseteq U$ and $D_n\cap D_k=\varnothing$ for $k<n$.
Now let $$B^*=B\cup \bigcup _{n\in\omega} D_n.$$ 
Then $B^*$ is a bubble set and $B\subseteq B^*$. 
$B^*$ is dense in $P$:
(i) $\{p_i:i\in\omega\}\subseteq \bigcup _{n\in\omega} \overline D_n$ implies that $\overline U= \overline{\{p_i:i\in\omega\}}\subseteq  \overline {\bigcup_{n\in\omega}D_n}\subseteq \overline {B^*}$.
(ii) $P\setminus \overline U=B\subseteq B^*$.
