Definition:Boundary (Topology)/Definition 2

Definition

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

Let $H \subseteq S$.

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$x \in S$ 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 $H$.

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

• 1971: William W. Fairchild and Cassius Ionescu Tulcea: Topology: $3$: Interior of a Set and Adherence of a Set, Definition $3.22$