TITLE: The boundary of set is the set itself examples
QUESTION [5 upvotes]: Could you provide me some examples of sets, which are not based on Cantor's construction, that satisfy the property $\partial A=A$, that is the boundary of a set is the set itself?

REPLY [11 votes]: Any closed set with an empty interior is its own boundary. @MichaelHardy gave you two examples in his comment: the circle $x^2 + y^2 = 1$ in the plane and the set of $\mathbb{Z}$ of all integers within the line $\mathbb{R}$. Also @kccu noted that you can take any subset of $\mathbb{R}^n$ viewed as a subset of $\mathbb{R}^m$ for $m > n$, like a line in the plane, or a surface in $\mathbb{R}^3$.