Boundary Point

Topics: Calculus - Topology

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(definition)

Let $A\subseteq {\mathbb{R}}^n$, $x\in {\mathbb{R}}^n$. We say that $x$ is a boundary point if $\forall r>0$:

$$B_r(x)\cap A\ne \varnothing$$

$$B_r(x)\cap A^c\ne \varnothing$$

We denote the set of boundary points with $\partial A$.

Examples

• $\partial (0,1)=\{0,1\}$
• $\partial [0,1]\times [0,1]=\left\{(x,y)\in {\mathbb{R}}^2|\begin{array}{c}x=0,0\le y\le 1\\ x=1,0\le y\le 1\\ y=0,0\le x\le 1\\ y=1,0\le x\le 1\end{array}\right\}$