I'm currently taking a class on multivariable calculus. The textbook begins by defining interior, exterior and boundary points in $\mathbb{R}^n$ and states that, given a set $M\subset\mathbb{R}^n$, every point $\mathbf{a}\in\mathbb{R}^n$ is either an interior, exterior or boundary point of $M$.
I recognize this must be true for $n = 1, 2, 3$ for geometric reasons but fail to see why it holds in $\mathbb{R}^n$ where $n>3$. I tried proving it using the definitions of interior, exterior and boundary points in $\mathbb{R}^n$ but got nowhere.