Definition:Boundary (Topology)/Definition 1

Definition

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

Let $H \subseteq S$.

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 $H$.

Also known as

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

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

• Equivalence of Definitions of Boundary

• Boundary is Intersection of Closure with Closure of Complement

• Results about set boundaries can be found here.

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction: Closures and Interiors