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?
The answer is negative. Consider two full closed disks touching in one point in the euclidean plane, with induced topology. If you take the interior, you get the disjoint union of two open disks, which is not connected anymore.