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

Question

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

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I know that if $C$ is connected then $\overline{C}$ is connected but should be the case here

Answer 1

for the second part consider- the open intervals.

Answer 2

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