Subbases of a Topology

Recall from the Bases of a Topology page that if $(X, \tau)$ is a topological space then a base for the topology $\tau$ is a collection $\mathcal B$ of open sets such that every $U \in \tau$ can be written as a union of a subcollection of open sets from $\mathcal B$. In other words, for every $U \in \tau$ 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 define a similar term known as a subbase

Definition: Let $(X, \tau)$ be a topological space. A collection $\mathcal S \subseteq \tau$ is called a Subbase (sometimes Subbasis) for $\tau$ if the collection of finite intersections of elements from $\mathcal S$ forms a basis of $\tau$, i.e. $\mathcal B_S = \{ U_1 \cap U_2 \cap ... \cap U_k : U_1, U_2, ..., U_k \in \mathcal S \}$ is a basis of $\tau$.

The following proposition gives us an alternative definition of a subbase for a topology.

Proposition 1: Let $(X, \tau)$ be a topological space. Then $\mathcal S$ is a subbase for $\tau$ if and only if $\tau$ is the smallest topology containing $\mathcal S$.

Example 1

Consider the topological space $(\mathbb{R}, \tau)$ where $\tau$ is the usual topology of open intervals in $\mathbb{R}$. Consider the following set of semi-infinite open intervals:

(2)

\begin{align} \quad S = \{ (-\infty, b) : b \in \mathbb{R} \} \cup \{ (a, \infty) : a \in \mathbb{R} \} \cup \{ \mathbb{R} \} \end{align}

Notice that for $a, b \in \mathbb{R}$ and $a < b$ we have that:

(3)

\begin{align} \quad (-\infty, b) \cap (a, \infty) = (a, b) \end{align}

For $a \geq b$ we have that $(-\infty, b) \cap (a, \infty) = \emptyset$. Therefore the collection of all finite intersctions of elements from $\mathcal S$ are either open intervals or the empty set. Recall that the collection of open intervals already forms a basis of the usual topology on $\mathbb{R}$. Hence $S$ is a subbasis of $\tau$ since $\mathcal B_S = \{ U_1 \cap U_2 \cap ... \cap U_k : U_1, U_2, ..., U_k \}$ is a basis of $\tau$.