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

## Mathematics Subject Classification

54-00*no label found*

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