Prove or disprove that the interior and the boundary of a connected set are connected.

For the boundary as poton says open interval is a counter example. For interior consider union of closed disks of radius 1 centered at (-1,0) and (1,0) in the plane.

It is surely connected but its interior i.e. union of open disks is not connected

For the first part: Let $X=\{a,b,c,d\}$ and the topology on $X$ is $\tau=\{X,\phi,\{a\},\{b\},\{a,b\},\{a,b,d\}\}$. Let $A=\{a,b,c\}$ the set $A$ is connected but the interior of it is $\mathrm{int}(A)=\{a,b\}$ is disconnected.

for the second part consider- the open intervals.