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
\[\partial A = \overline{A} \setminus \operatorname{int}(A).\]Examples
Worked examples
-
The closed unit disk $D = \setdef{(x, y) \suchthat x^2 + y^2 \leq 1}$
: $\partial D = \setdef{(x, y) \suchthat x^2 + y^2 = 1}$
High school
The boundary of the closed unit disk $D$ is the unit circle.
-
$[0, 1]$
: $\partial [0, 1] = \setdef{0, 1}$
High school
The boundary of $[0, 1]$ is $\setdef{0, 1}$, because every interval around $0$ or $1$ contains points inside and outside $[0, 1]$.
-
$[0, 1)$, as a subset of $\reals$
: $\partial [0, 1) = \setdef{0, 1}$
High school
The boundary contains $1$ even though the set does not, and $0$ even though the set does.
-
$(0, 1)$
: $\partial (0, 1) = \setdef{0, 1}$
High school
The boundary of $(0, 1)$ is $\setdef{0, 1}$, because every interval around $0$ or $1$ contains points inside and outside $(0, 1)$.
-
$\mathbb{Q}$, as a subset of $\reals$
: $\partial \mathbb{Q} = \reals$
Undergraduate
Every interval contains both rational and irrational numbers, so every real number is a boundary point.
Examples
-
The unit circle, for the closed unit disk in $\reals^2$
High school
The unit circle is the boundary of the closed unit disk because every ball around a point of the circle contains points inside and outside the disk.
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: $\partial A \cap A = \emptyset$.