Corollary

Let $U$ be an open subset of a locally connected space $X$, and let $V$ be a component of $U$. Then the frontier of $V$ (relative to $X$) is contained in the frontier of $U$.

For $V$ is open and closed in $U$, hence a frontier point of $V$ (relative to $X$) cannot belong to $U$, for it would also be a frontier point of $V$ relative to $U$, and there is none.