boundary / frontier

Definition.Let $X$ be a topological space and let $A$ be a subsetof $X$. The boundary (or frontier) of $A$ is the set$\partial A=\overline{A}\cap \overline{X\backslash \backslash A}$,where the overline denotes the closure of a set.Instead of $\partial A$, many authors use some other notationsuch as $\mathrm{bd}(A)$, $\mathrm{fr}(A)$, $A^b$ 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 $\partial A=\partial (X\backslash \backslash A)$,and $\partial X=\mathrm{\varnothing }=\partial \mathrm{\varnothing }$.

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 $\{0,1\}$.