Let $C$ be a connected component of $X\subset\mathbb{R}^n$. I want to prove of disprove that $\partial{C}\subset\partial{X}$ (where $\partial{A}$ means the boundary set of $A$).
In metric space, I know a connected component is closed.
In locally connected space(clearly, $\mathbb{R}^n$ is locally connected), I know a connected component is open.
So, in $\mathbb{R}^n$, any components are clopen. Therefore, the boundary set of a given connected component $C$ is exactly the empty set, which trivially is contained in $\partial{X}$
However, it looks very strange to me. There must be something wrong.
I know this question is definitely a newbie topology question. Any explanation will be appreciated.