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.

Also see

• Results about set boundaries can be found here.

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$: Closures and Interiors