If $x\in \mathbb{C}$ and $r>0$, denote by $B(x,r)$ the open ball in $\mathbb{C}$ with center $x$ and radius $r$.
Suppose that $A\subset B(0, \rho)$ is compact, and that $A_{0}$ is a connected component of $0$ in $A$.
From this, Can I conclude that there exists a compact subset $K$ that contains $A_{0}$ and with $0\in\mbox{int}(K)$ such that $\partial K \cap A=\emptyset$?
It started like this:
From the compacity of $A_{0}$ it follows that there exists $\delta>0$ and $a_{1}, \cdots, a_{n}\in A_{0}$ such that $A_{0}\subseteq \bigcup_{i=1}^{n}B(a_{i}, \delta)$, my candidate for $K$ could be $\bigcup_{i=1}^{n}\overline{B(a_{i}, \delta)}$. But,I don't know what I can conclude from there.
About open compact connected subsets we have the following theorem:
Theorem (Sura-Bura) : Every compact component $A$ of a locally compact (Hausdorff) space $X$ has a neighborhood base in $X$ consisting of open compact susbsets of $X$.