Connected components of a close subset or a open subset of a topological space

Question

Whether the connected components of a close subset and those of an open subset of topological space are still close and open respectively?

The first answer is yes, while the second is not generally right. The first we can simply use the fact that the closure of connected set is still connected. The second, it is right for a locally connected topological space. A space $X$ is locally connected if and only if every component of every open set of $X$ is open?

Sorry I find the answer in the course of writing this question.

Answer 1

Indeed components of a space $X$ are closed in $X$, because the closure of a connected set is also connected. So if $X \subseteq Y$ is closed the components of $X$ are closed in $X$ and so closed in $Y$ (closed in closed is closed).

A classical theorem says that all connected components of all open subsets of $X$ are open (in that open subset or equivalently in $X$) iff $X$ is locally connected.