Find set $A$ whose limit points are the same as its boundary points

Question

I was reading a problem that asked to find a non-empty set $A$ such that set of limit points in $A$ is the same as the set of boundary points in $A$. The solution given was to pick the set of all $x\in\mathbb{R^2}$ such that $|x|=1$. I was wondering if there are any other such sets $A$? Below is the problem and solution in case you wanted to see the solution they had.

Edit Also, I'm a bit confused by their solution. If we were to consider the entire circle, the boundary of the circle would be the outer points. but since we are only considering those outer points, what are they bounding?

Thanks

Answer 1

Almost any curve in $\mathbb{R}^2$ will have this property: any conic section, graph of an elementary function, etc. (Every point in the curve is a limit point of the curve, and of its complement)

Answer 2

Another less trivial example is the set of rational numbers.

The set of its boundary points is the set of real numbers which is also the set of its limit points.