Definition:Boundary (Topology)/Definition 3

Definition

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

Let $H \subseteq S$.

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 $H$ 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

• Results about set boundaries can be found here.

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.7$: Definitions: Definition $3.7.31$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): frontier (boundary)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): frontier (boundary)