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 $H^b$.


Also see

  • Results about set boundaries can be found here.


Sources