Definition:Boundary (Topology)

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 $H$.

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:
 * $\partial H$
 * $\operatorname{b} \left({H}\right)$
 * $\operatorname{fr} \left({H}\right)$ (where $\operatorname{fr}$ stands for frontier)
 * $H^b$

The notations of choice on this website are $\partial H$ and $H^b$.

Also see

 * Boundary is Intersection of Closure with Closure of Complement


 * Boundary (geometric definition)