Definition:Boundary (Topology)/Definition 2

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

Let $H \subseteq S$.

We say that $x \in S$ is a boundary point of $H$ if every neighborhood $N$ of $x$ satisfies:
 * $H \cap N \ne \O$

and
 * $\paren{S \setminus H} \cap N \ne \O$

The boundary of $H$ consists of all the boundary points of $H$.

Also known as
The boundary of a subset $H$ is also seen referred to as the frontier of $H$.

Also see

 * Boundary is Intersection of Closure with Closure of Complement