Bases of a Topology Examples 2

Bases of a Topology Examples 2

Recall from the Bases of a Topology page that if $(X, \tau)$ is a topological space then a basis for the topology $\tau$ is a collection $\mathcal B \subseteq \tau$ such that every $U \in \tau$ can be written as a union of elements from $\mathcal B$, i.e., for all $U \in \tau$ we have that there exists a $\mathcal B^* \subseteq \mathcal B$ such that:

(1)
\begin{align} \quad U = \bigcup_{B \in \mathcal B^*} B \end{align}

We will now look at some more examples of bases for topologies.

Example 1

Consider the topological space $(\mathbb{R}, \tau)$ where $\tau$ is the usual topology of open intervals on $\mathbb{R}$. Show that $\mathcal B = \left \{ (p, q) : p, q \in \mathbb{Q}, p < q \right \}$ forms a basis of $\tau$.

It suffices to show that every open interval $(a, b) \in \tau$ can be expressed as a union of basis elements since every open set in $\tau$ is simply a union of open intervals. Let $(a, b) \in \tau$.

If $a, b \in \mathbb{Q}$ then $(a, b) \in \mathcal B$ and we're done.

If $a \in \mathbb{Q}$ and $b \not \in \mathbb{Q}$ then:

(2)
\begin{align} \quad (a, b) = \bigcup_{a < q < b}_{q \in \mathbb{Q}} (a, q) \end{align}

Similarly, if $a \not \in \mathbb{Q}$ and $b \in \mathbb{Q}$ then:

(3)
\begin{align} \quad (a, b) = \bigcup_{a < p < b}_{p \in \mathbb{Q}} (p, b) \end{align}

The last case occurs when $p, q \not \in \mathbb{Q}$. Then:

(4)
\begin{align} \quad (a, b) = \bigcup_{a < p < q < b}_{p, q \in \mathbb{Q}} (p, q) \end{align}

So for every open interval $(a, b)$ there exists a $\mathcal B^* \subseteq \mathcal B$ such that $(a, b) = \bigcup_{B \in \mathcal B^*} B$, so $\mathcal B$ is a basis of $\tau$.

Example 2

Consider the topological space $(\mathbb{R}, \tau)$ where $\tau$ is the usual topology of open intervals in $\mathbb{R}$. Show that the collection $\mathcal B = \{ (w, z) : w, z \in \mathbb{Z}, w < z\}$ does not form a basis of $\tau$.

If $\mathcal B$ was a basis of $\tau$ then for every set $U \in \tau$ we must be able to find a subcollection $\mathcal B^* \subseteq \mathcal B$ such that:

(5)
\begin{align} \quad U = \bigcup_{B \in \mathcal B^*} B \end{align}

Consider the open interval $U = \left ( 0, \frac{1}{2} \right )$. Any set in $\mathcal B$ that contains $U$ must be such that $w \leq 0$ and $\frac{1}{2} \leq z$. The smallest integer greater than $\frac{1}{2}$ is $z = 1$, and so the smallest set in $\mathcal B$ containing $U$ is $(0, 1)$.

However, there are no sets in $\mathcal B$ that are contained in $\left (0, \frac{1}{2} \right )$ and so $\mathcal B$ cannot be a basis of $\tau$.

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