Definition:Boundary (Topology)

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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 $T$:

$\partial H = H^- \cap \paren {\overline H}^-$


Also known as

The boundary of a subset $H$ is also seen referred to as the frontier of $H$.


Also defined as

Some sources define the boundary of a subset as:

the set of elements of $S$ which can be approached from within $S$ and from outside $S$.

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.


Sources