Boundary of an open set

#boundary#open set#disjointness

Statement

Show that an open set $U\subseteq\mathbb{R}^n$ is disjoint from its boundary.

Hint

Boundary $=\overline U\setminus\mathring U$ .

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Compute. Since $U$ is open, $\mathring U=U$ . Thus $\partial U=\overline U\setminus U$ .

Disjointness. $\partial U\cap U=\emptyset$ .

Conclusion. Open sets do not meet their boundary.