Boundary of a set
Elementary
Consider a shape drawn on a piece of paper using a continuous line that forms a closed loop. There is a portion of the paper that is inside the shape and a portion that is outside. The boundary of the shape is curve that separates the inside from the outside.
Undergraduate (subsets of $\reals^n$)
Consider a set $A \subseteq \reals^n$ and a point $x$. The point $x$ is in the boundary of $A$ if every open ball around $x$ contains both a point of $A$ and a point not in $A$.
Graduate (topological spaces)
Consider a subset $A$ of a topological space, with its closure $\overline{A}$ and its interior $\operatorname{int}(A)$. The boundary of $A$ is $\partial A = \overline{A} \setminus \operatorname{int}(A)$.
Properties
- $\partial A = \overline{A} \cap \overline{X \setminus A}$, so a set and its complement have the same boundary.
- The boundary of every set is closed.
- A set $A$ is closed if and only if it contains its boundary: $\partial A \subset A$,
- A set $A$ is open if and only if it is disjoint from its boundary: $\partial A \cap A = \emptyset$.