Definition:Boundary (Topology)

Let $$T$$ be a topological space.

Let $$X \subseteq T$$.

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");
 * $$\partial X$$.

Thus we can write $$\partial X = \operatorname{cl} \left({X}\right) - \operatorname{Int} \left({X}\right)$$.

Alternatively, from Complement of Closure is Interior of Complement, $$\partial X = \operatorname{cl} \left({X}\right) \cap \operatorname{cl} \left({T - X}\right)$$.

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