Definition:Boundary (Topology)

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This page is about boundaries in the context of topology. For other uses, see Definition:Boundary.

Definition

Let $T = \left({S, \tau}\right)$ be a topological space.

Let $H \subseteq S$.


Then 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 $H$ is also seen referred to as the frontier of $H$.


Notation

The boundary of $H$ is variously denoted (with or without the brackets):

$\partial H$
$\map {\operatorname b} H$
$\map {\operatorname {Bd} } H$
$\map {\operatorname {fr} } H$ (where $\operatorname {fr}$ 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