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 of $$X$$ consists of all the points in the closure of $$X$$ which are not in the interior of $$X$$.