Definition:Boundary (Geometry)

Geometry
As defined by Euclid:

"A boundary is that which is an extremity of anything."

For example, the endpoints of a line segment are its boundaries.

Containment
A figure is said to be contained by its boundary or boundaries.

Topology
Let $$T$$ be a topological space.

Let $$X \subseteq T$$.

Then the boundary $$b \left({X}\right) = \operatorname{cl} \left({X}\right) - \operatorname{Int} \left({X}\right)$$.

That is, the boundary of $$X$$ consists of all the points in the closure of $$X$$ which are not in the interior of $$X$$.

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

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