Definition:Boundary (Topology)

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

Let $X \subseteq S$.

Then the boundary of $X$ (or frontier of $X$) consists of all the points in the closure of $X$ which are not in the interior of $X$.

The boundary of $X$ is variously denoted:
 * $\operatorname{b} \left({X}\right)$
 * $\operatorname{fr} \left({X}\right)$ (where $\operatorname{fr}$ stands for frontier)
 * $\partial X$
 * $X^b$

Thus we can write:
 * $\partial X = \operatorname{cl} \left({X}\right) \setminus \operatorname{Int} \left({X}\right)$

or:
 * $\partial X = X^- \setminus X^\circ$

using $X^-$ for closure and $X^\circ$ for interior of $X$.

Alternatively, from Boundary is Intersection of Closure with Closure of Complement:
 * $\partial X = \operatorname{cl} \left({X}\right) \cap \operatorname{cl} \left({S \setminus X}\right)$

or:
 * $\partial X = X^- \cap \left({S \setminus X}\right)^-$

Note
It can be intuitively perceived that the topological and geometric definitions of boundary are compatible.