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

Also see

 * Equivalence of Definitions of Boundary


 * Boundary is Intersection of Closure with Closure of Complement