Is the following statement is true/false ?
If a set $A$ is connected, then so interior $A^o$ is also connected.
My attempt : I think this statement is true I have not found any contradiction I was thinking about $A = [0,1]$, or $\mathbb{R}$
2026-04-09 10:11:44.1775729504
True/false :If a set $A$ is connected, then so interior $A^o$ is also connected
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This is not true. Consider union of two closed balls in $\mathbb R^{2}$ touching each other at one point. The interior of this set is the union of the corresponding open balls. These two open balls are non-empty disjoint open sets and their union is not connected by the very definition of connectedness.