The Open and Closed Sets of a Topological Space Examples 1

Recall from The Open and Closed Sets of a Topological Space page that if $(X, \tau)$ is a topological space then a set $A \subseteq X$ is said to be open if $A \in \tau$ and $A$ is said to be closed if $A^c \in \tau$. Furthermore, if $A$ is both open and closed, then we say that $A$ is clopen.

We will now look at some examples of identifying the open, closed, and clopen sets of a topological space $(X, \tau)$.

Example 1

Let $X = \{ a, b, c, d \}$ and consider the topology $\tau = \{ \emptyset, \{ c \}, \{ a, b \}, \{ c, d \}, \{a, b, c \}, X \}$. What are the open, closed, and clopen sets of $X$ with respect to this topology?

The open sets of $X$ are those sets forming $\tau$:

(1)

\begin{align} \quad \mathrm{open \: sets \: of \: X} = \{ \emptyset, \{ c \}, \{ a, b\} , \{c, d \}, \{ a, b, c \}, X \} \end{align}

The closed sets of $X$ are the complements of all of the open sets:

(2)

\begin{align} \quad \mathrm{closed \: sets \: of \: X } = \{ \emptyset, \{a, b, d \}, \{c, d \}, \{a, b \}, \{ d \}, X \} \end{align}

The clopen sets of $X$ are the sets that are both open and closed:

(3)

\begin{align} \quad \mathrm{clopen \: sets \: of \: X} = \{ \emptyset, \{a, b \}, \{c, d \}, X \} \end{align}

Example 2

Prove that if $X$ is a set and every $A \subseteq X$ is clopen with respect to the topology $\tau$ then $\tau$ is the discrete topology on $X$.

Let $X$ be a set and let every $A \subseteq X$ be clopen. Then every $A \subseteq X$ is open, i.e., every subset of $X$ is open, so $\tau = \mathcal P(X)$. Hence $\tau$ is the discrete topology on $X$.

Example 3

Consider the topological space $(\mathbb{Z}, \tau)$ where $\tau$ is the cofinite topology. Determine whether the set of even integers is open, closed, and/or clopen. Determine whether the set $\mathbb{Z} \setminus \{1, 2, 3 \}$ is open, closed, and/or clopen. Determine whether the set $\{-1, 0, 1 \}$ is open, closed, and/or clopen. Show that any nontrivial subset of $\mathbb{Z}$ is never clopen.

Recall that the cofinite topology $\tau$ is described by:

(4)

\begin{align} \quad \tau = \{ U \subseteq X : U = \emptyset \: \mathrm{or} \: U^c \: \mathrm{is \: finite} \} \end{align}

We first consider the set of even integers which we denote by $E = \{ ..., -2, 0, 2, ... \}$. We see that $E^c$ is the set of odd integers, i.e., $E^c = \{ ..., -3, -1, 1, 3, ... \}$ which is an infinite set. Therefore $E \not \in \tau$ so $E$ is not open. Furthermore, we have that $(E^c)^c = E$ is an infinite set and $E^c \not \in \tau$ so $E$ is not closed either.

We now consider the set $\mathbb{Z} \setminus \{1, 2, 3 \}$. We have that $(\mathbb{Z} \setminus \{1, 2, 3 \})^c = \{1, 2, 3 \}$ which is a finite set. Therefore $\mathbb{Z} \setminus \{1, 2, 3 \} \in \tau$, so $\mathbb{Z} \setminus \{1, 2, 3 \}$ is open. Now consider the complement $(\mathbb{Z} \setminus \{1, 2, 3 \})^c = \{1, 2, 3 \}$. The complement of this set is $(\{1, 2, 3 \})^c = \mathbb{Z} \setminus \{1, 2, 3 \}$ which is an infinite set, so $(\{ 1, 2, 3 \})^c \not \in \tau$. Hence $\mathbb{Z} \setminus \{1, 2, 3 \}$ is not closed.

Lastly we consider the set $\{ -1, 0, 1 \}$. We have that $(\{ -1, 0, 1 \})^c = \mathbb{Z} \setminus \{-1, 0, 1 \}$ which is an infinite set, so $\{-1, 0, 1 \} \not \in \tau$ so $\{ -1, 0, 1 \}$ is not open. Now consider the complement $(\{-1, 0, 1\})^c = \mathbb{Z} \setminus \{-1, 0, 1 \}$. The complement of this set is $\{ -1, 0, 1 \}$ is finite, seo $(\{-1, 0, 1\})^c \in \tau$. Hence $\{-1, 0, 1 \}$ is closed.

Lastly, let $A \subseteq \mathbb{Z}$ be a nontrivial subset of $\mathbb{Z}$, i.e., $A \neq \emptyset$ and $A \neq \mathbb{Z}$. Suppose that $A$ is clopen. Then $A$ is both open and closed. Hence by definition, $A$ and $A^c$ are both open. Hence $A^c$ and $A$ are both finite. But $\mathbb{Z} = A \cap A^c$ which implies that $\mathbb{Z}$ is a finite set - which is preposterous since the set of integers is an infinite set! Hence $A$ cannot be clopen.