The Topology of Closed Intervals [-n, n] on the Set of Real Numbers

# The Topology of Closed Intervals [-n, n] on the Set of Real Numbers

Recall from the Topological Spaces page that a set $X$ an a collection $\tau$ of subsets of $X$ together denoted $(X, \tau)$ is called a topological space if:

• $\emptyset \in \tau$ and $X \in \tau$, i.e., the empty set and the whole set are contained in $\tau$.
• If $U_i \in \tau$ for all $i \in I$ where $I$ is some index set then $\displaystyle{\bigcup_{i \in I} U_i \in \tau}$, i.e., for any arbitrary collection of subsets from $\tau$, their union is contained in $\tau$.
• If $U_1, U_2, ..., U_n \in \tau$ then $\displaystyle{\bigcap_{i=1}^{n} U_i \in \tau}$, i.e., for any finite collection of subsets from $\tau$, their intersection is contained in $\tau$.

On The Topology of Open Intervals on the Set of Real Numbers page we saw that if $\tau = \emptyset \cup \mathbb{R} \cup \{ (-n, n) : n \in \mathbb{Z}, n \geq 1 \}$ then $(X, \tau)$ is a topological space.

We will now look at a similar topology of closed intervals of the form $[-n, n]$ with $\emptyset$, $\mathbb{R}$ included on the set of real numbers.

Consider the following collection, from $\mathbb{R}$, of closed intervals with $\emptyset$ and $\mathbb{R}$ included:

(1)
\begin{align} \quad \tau = \{ \emptyset, \mathbb{R} \} \cup \{ [-n, n] : n \in \mathbb{Z}, n \geq 1 \} = \{ \emptyset, \mathbb{R}, [-1, 1], [-2, 2], ..., [-n, n], ... \} \end{align}

For the first condition, clearly $\emptyset, \mathbb{R} \in \tau$ by the definition of $\tau$.

For the second condition, notice that:

(2)
\begin{align} \quad \emptyset \subset [-1, 1] \subset [-2, 2] \subset ... \subset [-n, n] \subset ... \subset \mathbb{R} \end{align}

Therefore any arbitrary union $\displaystyle{\bigcup_{i \in I} U_i}$ for $U_i \in \tau$ for all $i \in I$ is the "largest" subset in the union and is hence contained in $\tau$.

For the third condition, we have that any finite intersection $\displaystyle{\bigcap_{i=1}^{n} U_i}$ for $U_i \in \tau$ and $i \in \{ 1, 2, ..., n \}$ is the "smallest subset in the intersection and is hence contaiend in $\tau$.

Therefore $(X, \tau)$ is a topological space.