Boundary

Definition

Let $X$ be a topological space and let $S \subseteq X$. The boundary of $S$ is the set difference $$\partial S = \overline{S} \setminus \operatorname{int} S,$$ where $\overline{S}$ is the closure and $\operatorname{int} S$ is the interior of $S$.

Remarks

A point $x$ is in $\partial S$ if and only if each open set that contains $x$ meets both $S$ and $X \setminus S$. The boundary is closed, and $\partial S = \partial (X \setminus S)$. In $\mathbb{R}$, the boundary of $(0, 1)$ is $\{0, 1\}$, and the boundary of $\mathbb{Q}$ is $\mathbb{R}$. Another name for the boundary is the frontier.

Explanations

L1 human

The boundary is the edge of a set. It has the points where you can step into the set and also out of it with a very small step. For a disc in the plane, the boundary is its circle.