Let $X$ a connected metric space. Is $int X$ connected?
I have a proposition that says: if $X\subseteq Y\subseteq \overline{X}$ and $X$ is connected then $Y$ is connected. But I'm having some problems with using this result. Because every set is open in itself we would have $X = intX \subseteq\overline{X}$ then $intX$ would be connected. Is this correct? What if $X$ is a connected subset of a (not necessarely connected) metric space?