# boundary / frontier

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 $\operatorname{bd}(A)$, $\operatorname{fr}(A)$, $A^{b}$ 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 $\mathbb{R}^{n}$. (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 $\{0,1\}$.

Title boundary / frontier BoundaryFrontier 2013-03-22 13:34:46 2013-03-22 13:34:46 yark (2760) yark (2760) 17 yark (2760) Definition msc 54-00 boundary frontier topological boundary ExtendedBoundary Interior