54-00-BoundaryFrontier.tex

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{\bf Definition.}
Let $X$ be a topological space and let $A$ be a subset 
of $X$. The \emph{boundary} (or \emph{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 $\{0,1\}$.
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