Definition. Let $X$ be a topological space and let $A$ be a subset of $X$ The boundary (or frontier) of $A$ is the set $\partial A = \overline{A}\cap \overline{X\backslash A}$ where the overline denotes the closure of a set. Instead of $\partial A$ many authors use some other notation such as $\bd(A)$ $\fr(A)$ $A^b$ or $\beta(A)$ Note that the $\partial$ symbol is also used for other meanings of `boundary'.

From the definition, it follows that the boundary of any set is a closed set. It also follows that $\partial A = \partial(X\backslash A)$ and $\partial X=\varnothing=\partial\varnothing$

The term `boundary' (but not `frontier') is used in a different sense for topological manifolds: the boundary $\partial M$ of a topological $n$ manifold $M$ is the set of points in $M$ that do not have a neighbourhood homeomorphic to $\R^n$ (Some authors define topological manifolds in such a way that they necessarily have empty boundary.) For example, the boundary of the topological $1$ manifold $[0,1]$ is $\{0,1\}$