Let $A$ be a topological space, $C$ be a connected component of $\bar{A}$, $\{A_i\}$ be the collection of connected components of $A$ contained in $C$. Is $\bigcup \bar{A_i}$ contained, strictly contained, equal or not contained in $C$?
Attempt. My guess is that if $A_i$ is a connected component of $A$, then $\bar{A_i}$ is contained in a connected component of $\bar{A}$. Hence the answer to the above question is "contained". Any help is deeply appreciated.
A connected component is closed, so $C$ is closed in $\bar A$, $A_i\subset C$ implies that $\bar A_i\subset\bar C=C$. This implies that $\cup_i\bar A_i\subset C$.