Clopen Set Criterion for Disconnected Topological Spaces

# Clopen Set Criterion for Disconnected Topological Spaces

Recall from the Connected and Disconnected Topological Spaces page that a topological space $X$ is said to be disconnected if there exists open sets $A, B \subset X$, $A, B \neq \emptyset$, $A \cap B = \emptyset$, and such that:

(1)
\begin{align} \quad X = A \cup B \end{align}

Furthermore we said that $\{ A, B \}$ is a separation of $X$. We also defined $X$ to be connected if $X$ is not disconnected.

We will now look at a nice criterion for determining whether a topological space is connected or disconnected with regards to clopen sets in the topological space of interest.

 Theorem 1: A topological space $X$ is disconnected if and only if there exists a clopen set (open and closed set) $A \subset X$ such that $A \neq \emptyset$ and $A \neq X$.
• Proof: $\Rightarrow$ Suppose that $X$ is disconnected. Then there exists open sets $A, B \subset X$, $A, B \neq \emptyset$, $A \cap B = \emptyset$, and $X = A \cup B$. Since $A \cap B = \emptyset$ and $X = A \cup B$ we see that $A^c = B$. So $A^c$ is an open set in $X$ which implies that $A$ is also closed in $X$. Similarly, $B$ is also a closed set in $X$ since $B = A^c$ and $A$ is open in $X$.
• Furthermore, the sets $A$ and $B$ are such that $A, B \neq \emptyset$ and $A, B \neq X$.
• $\Leftarrow$ Suppose that there exists a clopen set $A \subset X$ such that $A \neq \emptyset$ and $A \neq X$. Let $B = A^c$. We claim that $\{ A, B \}$ is a separation of $X$.
• Since $A$ is open (and closed), $B$ is also open (and closed). Furthermore, $A \neq \emptyset$. Since $A \neq X$ we also have that $B \neq \emptyset$. Obvious by definition $A \cap B = \emptyset$. Furthermore, by definition it is not hard to see that $X = A \cup B$.
• Hence $\{ A, B \}$ is a separation of $X$ which shows that $X$ is disconnected. $\blacksquare$

Sometimes the following (logically equivalent) corollary in terms of connected topological spaces is more useful to remember.

 Corollary 1: A topological space $X$ is connected if and only if the open clopen sets in $X$ are the empty set $\emptyset$ and the whole set $X$.
• Proof: Corollary 1 is logically equivalent to Theorem 1. $\blacksquare$