Definition 12.8.
Let \(A\) be a subset of a topological space \(X\text{.}\) A point \(a \in A\) is an interior point of \(A\) if \(A\) is a neighborhood of \(a\text{.}\)
Section The Interior of a Set in a Topological Space
We have seen that topologies define the open sets in a topological space. As in metric spaces, open sets can be characterized in terms of their interior points. We defined interior points in metric spaces in terms of neighborhoods — the same holds true in topological spaces.
Remember that a set is a neighborhood of a point if the set contains an open set that contains the point. By definition, every open set is a neighborhood of each of its points, so every point of an open set \(O\) is an interior point of \(O\text{.}\) Conversely, if every point of a set \(O\) is an interior point, then \(O\) is a neighborhood of each of its points and is open. This argument is summarized in the next theorem.
Theorem 12.9.
Let \(X\) be a topological space. A subset \(O\) of \(X\) is open if and only if every point of \(O\) is an interior point of \(O\text{.}\)
The collection of interior points in a set form a subset of that set, called the interior of the set.
Definition 12.10.
The interior of a subset \(A\) of a topological space \(X\) is the set
\begin{equation*}
\Int(A) = \{a \in A \mid a \text{ is an interior point of } A\}\text{.}
\end{equation*}
Activity 12.9.
(a)
Consider \((\R, \tau)\text{,}\) where \(\tau\) is the standard topology (by standard in this situation, we mean the metric topology determined by the Euclidean metric). Let \(A=(-\infty,0)\cup (1,2]\cup \{3\}\) in \(\R\text{.}\) What is \(\Int(A)\text{?}\) What is the largest open subset of \(\R\) contained in \(A\text{?}\)
(b)
Consider \((\R, \tau)\text{,}\) where \(\tau\) is the discrete topology (the one where all subsets are open). Let \(A=(-\infty,0) \cup (1,2] \cup \{3\}\) in \(\R\text{.}\) What is \(\Int(A)\text{?}\) What is the largest open subset of \(\R\) contained in \(A\text{?}\)
(c)
Consider \((\R, \tau)\text{,}\) where \(\tau\) is the finite complement topology (the one where the open sets are the empty set along with all subsets \(O\) of \(\R\) such that \(\R \setminus O\) is finite). Let \(A=(-\infty,0) \cup (1,2] \cup \{3\}\) in \(\R\text{.}\) What is \(\Int(A)\text{?}\) What is the largest open subset of \(\R\) contained in \(A\text{?}\)
(d)
Let \(X = \{a,b,c,d\}\) and let
\begin{equation*}
\tau = \{\emptyset, \{a\}, \{a,b\}, \{c\}, \{d\}, \{c,d\}, \{a,c,d\}, \{a,c\}, \{a,d\}, \{a,b,c\}, \{a,b,d\}, X\}\text{.}
\end{equation*}
Assume that \(\tau\) is a topology on \(X\text{.}\) Let \(A = \{b,c,d\}\text{.}\) What is \(\Int(A)\text{?}\) What is the largest open subset of \(X\) contained in \(A\text{?}\)
One might expect that the interior of a set is an open set, as it was in metric spaces. This is true, but we can say even more. In Activity 12.9 we saw that in our examples that \(\Int(A)\) was the largest open subset of \(X\) contained in \(A\text{.}\) That this is always true is the subject of the next theorem.
Theorem 12.11.
Let \((X,d)\) be a topological space, and let \(A\) be a subset of \(X\text{.}\) The interior of \(A\) is the largest open subset of \(X\) contained in \(A\text{.}\)
Proof.
Let \(X\) be a topological space, and let \(A\) be a subset of \(X\text{.}\) We need to prove that \(\Int(A)\) is an open set in \(X\text{,}\) and that \(\Int(A)\) is the largest open subset of \(X\) contained in \(A\text{.}\) First we demonstrate that \(\Int(A)\) is an open set. Let \(a \in \Int(A)\text{.}\) Then \(a\) is an interior point of \(A\text{,}\) so \(A\) is a neighborhood of \(a\text{.}\) This implies that there exists an open set \(O\) containing \(a\) so that \(O \subseteq A\text{.}\) But \(O\) is a neighborhood of each of its points, so every point in \(O\) is an interior point of \(A\text{.}\) It follows that \(O \subseteq \Int(A)\text{.}\) Thus, \(\Int(A)\) is a neighborhood of each of its points and, consequently, \(\Int(A)\) is an open set.
The proof that \(\Int(A)\) is the largest open subset of \(X\) contained in \(A\) is left for the next activity.
Activity 12.10.
Let \((X,d)\) be a topological space, and let \(A\) be a subset of \(X\text{.}\)
(a)
What will we have to show to prove that \(\Int(A)\) is the largest open subset of \(X\) contained in \(A\text{?}\)
(b)
Suppose that \(O\) is an open subset of \(X\) that is contained in \(A\text{,}\) and let \(x \in O\text{.}\) What does the fact that \(O\) is open tell us? Then complete the proof that \(O \subseteq \Int(A)\text{.}\)
One consequence of Theorem 12.11 is the following.
Corollary 12.12.
A subset \(O\) of a topological space \(X\) is open if and only if \(O = \Int(O)\text{.}\)
