Theorem 13.2.
Let \(X\) be a topological space.
- Any intersection of closed sets in \(X\) is a closed set in \(X\text{.}\)
- Any finite union of closed sets in \(X\) is a closed set in \(X\text{.}\)
Section Unions and Intersections of Closed Sets
Now we have defined open and closed sets in topological spaces. In our preview activity we saw that a set can be both open and closed. As we did in metric spaces, we will call any set that is both open and closed a clopen (for closed-open) set.
By definition, any union and any finite intersection of open sets in a topological space is open, so the fact that closed sets are complements of open sets implies the following theorem.
Proof.
Let \(X\) be a topological space. To prove part 1, assume that \(C_{\alpha}\) is a collection of closed set in \(X\) for \(\alpha\) in some indexing set \(I\text{.}\) Then
\begin{equation*}
X \setminus \bigcap_{\alpha \in I} C_{\alpha} = \bigcup_{\alpha \in I} X \setminus C_{\alpha}\text{.}
\end{equation*}
The latter is an arbitrary union of open sets and so it an open set. By definition, then, \(\bigcap_{\alpha \in I} C_{\alpha}\) is a closed set.
For part 2, assume that \(C_1\text{,}\) \(C_2\text{,}\) \(\ldots\text{,}\) \(C_n\) are closed sets in \(X\) for some \(n \in \Z^+\text{.}\) To show that \(C = \bigcap_{k=1}^n C_k\) is a closed set, we will show that \(X \setminus C\) is an open set. Now
\begin{equation*}
X \setminus \bigcup_{\alpha \in I} C_{\alpha} = \bigcap_{\alpha \in I} X \setminus C_{\alpha}
\end{equation*}
is a finite intersection of open sets, and so is an open set. Therefore, \(\bigcup_{\alpha \in I} C_{\alpha}\) is a closed set.
Theorem 13.2 tells us that any intersection of closed sets is closed, but only finite unions of closed sets are closed. How do we know that non-finite unions of closed sets aren’t necessarily closed?
Activity 13.2.
Let \(\Z\) be a topological space with the finite complement topology \(\tau_{FC}\text{.}\) That is, a non-empty set \(O\) is open in \(\Z\) if \(\Z \setminus O\) is finite.
(a)
What must be true about the cardinality of the closed sets in \((\Z, \tau_{FC})\text{?}\)
(b)
Let \(C_n = \{2, 3, \ldots, n\}\text{.}\) Is the set \(\bigcup_{n \geq 3} C_n\) a closed set in \((\Z, \tau_{FC})\text{?}\) Explain.
