Let $X$ be a Hausdorff, compact, connected space. Suppose that $A$ is a closed non-empty subset of $X$ such that $A\neq X$. Then how do we show that if $C$ is a connected component of $A$ the following is true: $C∩∂A≠∅$
I've tried to proceed by contradiction: if $C∩∂A=∅$, then I wanted to find a separation of $C$ to contradict that $C$ is connected, but I couldn't finish my idea.
I found the next proof, but I got lost in the last step:
Let $U=int(A)∩C$ and $V=int(A^c)∩C$. Both $U$ and $V$ are open in $C$ and $U∩V=∅$. Now $U∪V≠C$ because $C$ is connected. Finally, $∂A∩C=C∖(U∪V)$.
How is the last equation true?