How to show $\partial A = \varnothing \Rightarrow A=R^n$

Question

Let $A\subset R^n$ and dim$A=n$, $\partial A$ is the relative boundary of $A$. If $\partial A=\varnothing$ how to show $A$ is $R^n$ ?

Picture below is from XX page of Schneider R.-Convex Bodies_ The Brunn-Minkowski Theory-Cambridge University Press (2013).

Answer 1

As $\dim A=n$, then the relative boundary of course coincides with the usual boundary of $A$ as a subset of $\mathbb R^n$. Then $\mathbb R^n$ is the union of the interior of $A$ and the exterior of $A$, two open sets. As $\mathbb R^n$ is connected, one of the two is empty, and $A$ is not.