The Closure of a Set in a Topological Space Examples 2

Recall from The Closure of a Set in a Topological Space page that if $(X, \tau)$ is a topological space and $A \subseteq X$ then the closure of $A$ is the smallest closed set containing $A$.

We will now look at some more examples of the closure of a set

Example 1

Consider the topological space $(\mathbb{N}, \tau)$ where $\tau$ is the cofinite topology. Find the closure of $A = \{ 1, 2, 3 \}$ and the closure of $E = \{ 2n : n \in \mathbb{N} \} = \{2, 4, 6, ... \}$.

Recall that the cofinite topology is described to be:

(1)

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

Therefore all open sets must be infinite (although not all infinite sets are open with this topology and except for the $\emptyset$). Furthermore, these open sets have finite complements so the closed sets must be finite sets (except for the wholeset $\mathbb{N}$ itself).

Hence $A = \{ 1, 2, 3 \}$ is a closed set. To show this, look at $A^c = \{ 1, 2, 3 \}^C = \mathbb{N} \setminus \{ 1, 2, 3 \}$. Now $(A^c)^c = \{ 1, 2, 3 \}$ is finite, so $A^c \in \tau$ so $A^c$ is open and hence $A$ is closed. Therefore the smallest closed set containing $\{1, 2, 3 \}$ is itself, i.e.,:

(2)

\begin{align} \quad \bar{A} = \overline{ \{1, 2, 3 \}} = \{1, 2, 3 \} \end{align}

For the same reasoning, if $A$ is a finite collection of natural numbers, $A = \{ n_1, n_2, ..., n_p \}$ with $n_1, n_2, ..., n_p \in \mathbb{N}$ then $\bar{A} = \{ n_1, n_2, ..., n_p \}$.

Now the set of even natural numbers $E = \{ 2n : n \in \mathbb{N}$ is an infinite set - but all closed sets apart from the whole set are closed. Therefore the smallest closed set containing $E$ is the wholeset $\mathbb{N}$ so:

(3)

\begin{align} \quad \bar{E} = \mathbb{N} \end{align}

For the same reasoning, if $B$ is an infinite collection of natural numbers then $\bar{B} = \mathbb{N}$!

Example 2

Consider the topological space $(\mathbb{R}^3, \tau)$ where $\tau$ is the usual topology of open balls in $\mathbb{R}^3$. What is the closure of an open half cone, $C \subseteq \mathbb{R}^3$?

It should not be too surprising that the closure of $C$ is:

(4)

\begin{align} \quad \bar{C} = \mathrm{The \: corresponding \: closed \: half \: cone \: in \:} \mathbb{R}^3 \end{align}