Is the interior of a connected set in $\mathbb R^k$ connected?
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$\begingroup$No. If $X\subset\mathbb R^2$ is the union of two closed disks of radius $1$, one with center at $(1;0)$ and another with center at $(-1;0)$, then $X$ is connected but its interior is not.
$\endgroup$ 2 $\begingroup$Nope... Pick two tangent balls...
$\endgroup$ 1 $\begingroup$No. Let
$A_1=\{(x,y)|x\leq0,y\leq0\}$ (The third quadrant and the positive x-axis) and
$A_2=\{(x,y)|x\geq0,y\geq0\}$.(The first quadrant and the positive x-axis).
$A_1\cup A_2$ is connected. Their interior is the first and third quadrants.
Proof by visualization.
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