# Non totally disconnected compact perfect set

Let $K$ be a non-empty compact set of reals without isolated points. Considering the usual topology, is it possible that $K$ is not totally disconnected (i.e., the singletons are not the unique connected sets) and not regular closed (i.e., the closure of the interior of $K$ is not $K$ itself)?

How about $C\cup [1,2],$ where $C$ is the usual Cantor set?