Definition:Boundary (Topology)
This page is about Boundary in the context of topology. For other uses, see Boundary.
Definition
Let $T = \struct {S, \tau}$ be a topological space.
Let $H \subseteq S$.
Definition from Closure and Interior
The boundary of $H$ consists of all the points in the closure of $H$ which are not in the interior of $H$.
Thus, the boundary of $H$ is defined as:
- $\partial H := H^- \setminus H^\circ$
where $H^-$ denotes the closure and $H^\circ$ the interior of $H$.
Definition from Neighborhood
$x \in S$ is a boundary point of $H$ if every neighborhood $N$ of $x$ satisfies:
- $H \cap N \ne \O$
and
- $\overline H \cap N \ne \O$
where $\overline H$ is the complement of $H$ in $S$.
The boundary of $H$ consists of all the boundary points of $H$.
Definition from Intersection of Closure with Closure of Complement
The boundary of $H$ is the intersection of the closure of $H$ with the closure of the complement of $H$ in $T$:
- $\partial H = H^- \cap \paren {\overline H}^-$
Definition from Closure and Exterior
The boundary of $H$ consists of all the points in $H$ which are not in either the interior or exterior of $H$.
Thus, the boundary of $H$ is defined as:
- $\partial H := H \setminus \paren {H^\circ \cup H^e}$
where:
Also known as
The boundary of a subset $H$ of a topological space $T$ is also seen referred to as the frontier of $H$.
Also defined as
Some sources define the boundary of a subset of a topological space $T = \struct {S, \tau}$ as:
However, this definition is of little use without rigorous definitions of approach, inside and outside.
Consequently, such imprecise definitions are not used on $\mathsf{Pr} \infty \mathsf{fWiki}$.
Notation
The boundary of $H$ is variously denoted (with or without the brackets):
- $\partial H$
- $\map {\mathrm b} H$
- $\map {\mathrm {Bd} } H$
- $\map {\mathrm {fr} } H$ or $\map {\mathrm {Fr} } H$ (where $\mathrm {fr}$ stands for frontier)
- $H^b$
The notations of choice on $\mathsf{Pr} \infty \mathsf{fWiki}$ are $\partial H$ and $H^b$.
Examples
Half-Open Real Interval
Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.
Let $\hointl a b$ be a half-open interval of $\R$.
Then the boundary of $\hointl a b$ is the set of its endpoints $\set {a, b}$.
Open Unit Interval
Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.
Let $\openint 0 1$ be the open unit interval in $\R$.
Then the boundary of $\openint 0 1$ is the set of its endpoints $\set {0, 1}$.
$\Z$ in $\R$
Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.
Let $\Z$ be the set of integers.
Then the boundary of $\Z$ in $\struct {\R, \tau_d}$ is $\Z$ itself.
Reciprocals in $\R$
Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.
Let $S$ be the set defined as:
- $S = \set {\dfrac 1 n: n \in \Z_{>0} }$
Then the boundary of $S$ in $\struct {\R, \tau_d}$ is $S \cup \set 0$.
Rationals in Closed Unit Interval in $\R$
Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.
Let $S$ be the set defined as:
- $S = \Q \cap \closedint 0 1$
where:
- $\Q$ denotes the set of rational numbers
- $\closedint 0 1$ denotes the closed unit interval.
Then the boundary of $S$ in $\struct {\R, \tau_d}$ is $\closedint 0 1$.
Also see
- Results about set boundaries can be found here.