Definition:Boundary (Topology)/Definition 4

Definition

Let $T = \struct {S, \tau}$ be a topological space.

Let $H \subseteq S$.

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 $H$.

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

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 $H^b$.