As in the title i want to find a set $A$ from a topological space such that $$ A~\text{ is not connected but}~~ \rm int(A)~\text{ is connected} $$
i tryed with this space: $E=\{a,b,c,d,e\}$ with $$\tau=\{\emptyset, E,\{b\},\{c\},\{b,c\},\{c,d\},\{b,c,d\}\}$$
$E$ is connected, $\{b,c,e\}$ is connected but it's interior no, but i don't find a set not connected but this interior connected.