The Open and Closed Sets of a Topological Space Examples 2

The Open and Closed Sets of a Topological Space Examples 2

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

Consider the topological space $(\mathbb{R}, \tau)$ where $\tau = \{ (-n, n) : n \in \mathbb{Z}, n \geq 1 \}$. What is the largest open set contained in the interval $(-\pi, e)$? Show that every nontrivial subset $A \subseteq \mathbb{R}$ cannot be clopen.

The open sets of $\mathbb{R}$ with respect to the topology $\tau$ above are:

(1)
\begin{align} \quad \mathrm{open \: sets \: of \:} \mathbb{R} = \{ (-1, 1), (-2, 2), (-3, 3), ... \} \end{align}

Consider the interval $(-\pi, e) \approx (-3.14159..., 2.71828...)$. We can clearly see that $(-1, 1) \subset (-\pi, e)$ and $(-2, 2) \subset (-\pi, e)$, but $(-3, 3) \not \subset (-\pi, e)$ since, for example, $2.9 \in (-3, 3)$ and $2.9 \not \in (-\pi, e)$. We see that the open sets of $\mathbb{R}$ with respect to the topology $\tau$ are nested, i.e.:

(2)
\begin{align} \quad (-1, 1) \subset (-2, 2) \subset ... \subset (-n, n) \subset ... \end{align}

Therefore the largest open set contains in the interval $(-\pi, e)$ is $(-2, 2)$.

Now let $A \subseteq X$ be a nontrivial subset of $X$, i.e., $A \neq \emptyset$ and $A \neq X$. Suppose that $A$ is clopen. Then $A$ and $A^c$ are both open. If $A$ is open, then $A = (-i, i)$ for some $i \in \mathbb{R}$. But then $A^c = (-\infty, i] \cup [i, \infty) \not \in \tau$ which is a contradiction. Therefore $A$ cannot be clopen.

Example 2

Prove that if $(X, \tau)$ is a topological space where $\tau$ is a nested topology then every nontrivial subset $A \subseteq X$ cannot be clopen.

Let $(X, \tau)$ be a topological space and suppose that $\tau$ is a nested topology. Then for $\tau = \{ \emptyset, U_1, U_2, ..., U_n, ..., X \}$ we have that:

(3)
\begin{align} \quad \emptyset \subset U_1 \subset U_2 \subset ... \subset U_n \subset ... \subset X \end{align}

Now suppose that $A \subset X$ is a nontrivial subset of $X$. Then $A \neq \emptyset$ and $A \neq X$. Suppose that $A$ is clopen. Then both $A$ and $A^c$ are open. Hence, for some $m, n \in \mathbb{N}$ we have that $A = U_m$ and $A^c = U_n$. But $A \cap A^c = \emptyset$, so $U_m \cap U_n = \emptyset$. But this happens if and only if either $U_m = \emptyset$ or $U_n = \emptyset$ due to the nesting above. Therefore either $A = \emptyset$ or $A^c = \emptyset$ which implies that $A = \emptyset$ or $A = X$ - a contradiction. Hence every nontrivial subset $A \subseteq X$ cannot be clopen.

Example 3

Let $X$ be nonempty finite set and let $(X, \tau)$ be a topological space. Prove that the number of clopen sets of $X$ with respect to the topology $\tau$ is always even.

Let $m$ denote the number of clopen subsets of $X$ with respect to $\tau$. Then clearly $m \geq 2$ since the sets $\emptyset$ and $X$ are always clopen. Let $A$ be an arbitrary clopen set. Then $A$ and $A^c$ are open sets, so $A^c$ is also a clopen set. Hence, every clopen set comes as a complementary pair and $A \neq A^c$. Hence, the number of clopen sets with respect to the topology $\tau$ is always even provided that $X$ is a nonempty finite set.

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