connected and compact subset of $\mathbb{C}$ is either a singleton or has uncountable boundary Let's say that $A\subseteq\mathbb{C}$ is connected and compact but not a singleton.  I want to conclude that $\partial A$ is uncountable.  Can anyone help me find a reference to cite?
I can see how to prove this in a routine way, by picking distinct elements $a$ and $b$ in $A$, which we may assume without loss of generality have distinct real parts, and then finding an element in $\partial A$ with any real part strictly between the real parts of $a$ and $b$.  However, I would much rather just give a reference, since the argument is uninteresting.

The boundary of such a set is a perfect set (closed with no isolated points).  Such set in $\Bbb R^k$ is uncountable.  See Rudin's Principles of Mathematical Analysis Theorem 2.43.
