Section The Boundary of a Set
In addition to limit points, we also defined boundary points in metric spaces. Recall that a boundary point of a set \(A\) in a metric space \(X\) could be considered to be any point in \(\overline{A} \cap \overline{X \setminus A}\text{.}\) We make the same definition in a topological space.
Definition 13.10.
Let \((X, \tau)\) be a topological space, and let \(A\) be a subset of \(X\text{.}\) A boundary point of \(A\) is a point \(x \in X\) such that every neighborhood of \(x\) contains a point in \(A\) and a point in \(X \setminus A\text{.}\) The boundary of \(A\) is the set
\begin{equation*}
\Bdry(A) = \{x \in X \mid x \text{ is a boundary point of } A\}\text{.}
\end{equation*}
As with metric spaces, the boundary points of a set \(A\) are those points that are “between” \(A\) and its complement.
Activity 13.6.
Find the boundaries of the following sets
(a)
\(\{c,d\}\) in \((X, \tau)\) with \(X= \{a,b,c,d\}\) and \(\tau = \{\emptyset, \{a\}, \{b\}, \{a,b\}, X \}\text{.}\)
(b)
\(\{a,b\}\) in the set \(X= \{a,b,c,d,e,f\}\) with topology
\begin{equation*}
\tau= \{\emptyset,\{b\}, \{a,b,c\},\{d,e,f\},\{b,d,e,f\}, X\}\text{.}
\end{equation*}
(c)
\(\{a,b\} \subset X\) where \(X = \{a,b,c\}\) in the discrete topology.
(d)
\(\Z\) in \(\R\) with the finite complement topology \(\tau_{FC}\text{.}\)
Just as with metric spaces, we can characterize the closed sets as the sets that contain their boundary.
Theorem 13.11.
A subset \(C\) of a topological space \(X\) is closed if and only if \(C\) contains its boundary.
