TITLE: Prove or disprove that the interior and the boundary of a connected set are connected.
QUESTION [7 upvotes]: Prove or disprove that the interior and the boundary of a connected 
set are connected. 

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

REPLY [4 votes]: 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.