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

This definition needs to be completed. In particular: Boundary Point You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding or completing the definition. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{DefinitionWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

$x \in S$ is a boundary point of $H$ if and only 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 point 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:

$H^\circ$ denotes the interior of $H$

$H^e$ denotes the exterior of $H$.

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:

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

• Equivalence of Definitions of Boundary

• Boundary is Intersection of Closure with Closure of Complement

• Definition:Boundary (Geometry)

• Results about set boundaries can be found here.