Clopen Set Criterion for Disconnected Topological Spaces Examples 1

Recall from the Clopen Set Criterion for Disconnected Topological Spaces page that that a topological space $X$ is disconnected if and only if $X$ contains clopen set $A \subset X$, $A \neq \emptyset$.

We saw that if $A$ is a proper clopen subset of $X$ then $\{ A, A^c \} = \{ A, X \setminus A \}$ can form a separation for the set $X$.

Equivalently, we can say that a topological space $X$ is open if and only if $X$ does not contain any clopen sets.

We will now look at some examples concerning this very nice theorem.

Example 1

Let $X$ be a set containing more than $1$ element and give $X$ the discrete topology. Prove that $X$ is disconnected.

If $X$ has the discrete topology then every subset in $X$ is open in $X$. But then every subset of $X$ is also closed in $X$. Hence every subset of $X$ is clopen in $X$.

Since $X$ contains more than $1$ element, there exists a clopen set $A \subset X$, $A \neq \emptyset$. So $X$ is disconnected.

Example 2

Consider the set $X = \{ a, b, c, d \}$ with the topology $\tau = \{ \emptyset, \{ a\}, \{b \}, \{a, b \}, \{a, b, c \}, X \}$. Determine whether or not this set is connected.

The closed sets in $X$ with the topology $\tau$ are:

(1)

\begin{align} \quad \mathrm{Closed \: Sets} = \{ \emptyset, \{ d \}, \{c, d \}, \{a, c, d \} \{b, c, d \}, X \} \end{align}

So no subset of $X$ is both open and closed apart from the emptyset, $\emptyset$, and the whole set, $X$. So $X$ is connected.

Example 3

Consider the set $X = \{ a, b, c, d \}$. Give a non-discrete topology $\tau$ on $X$ that makes $X$ disconnected.

Consider the following topology:

(2)

\begin{align} \quad \tau = \{ \emptyset, \{a, b \}, \{c, d\}, X \} \end{align}

Then the closed sets of $X$ are also $\{ \emptyset, \{a, b \}, \{c, d \}, X \}$. So $\{a, b \}$ is a proper subset of $X$ that is both open and closed, so $X$ is disconnected and $\{ \{a, b \}, \{c, d \} \}$ form a separation of $X$.