Bases of Open Sets

We will now focus our attention at a special type of subset of a topology called a base for $\tau$ which we define below.

Definition: Let $(X,\tau )$ be a topological space. A Base (sometimes Basis) for the topology $\tau$ is a collection ${\mathcal {B}}$ of subsets from $\tau$ such that every $U\in \tau$ is the union of some collection of sets in ${\mathcal {B}}$.

Note that by definition, ${\mathcal {B}}=\tau$ is a base of $\tau$ - albeit a rather trivial one! The emptyset is also obtained by an empty union of sets from ${\mathcal {B}}$.

Let's look at some examples.

Example 1

Consider any nonempty set $X$ with the discrete topology $\tau ={\mathcal {P}}(X)$. Consider the collection:

${\begin{aligned}\quad {\mathcal {B}}=\{\{x\}:x\in X\}\end{aligned}}$

We claim that ${\mathcal {B}}$ is a base of the discrete topology $\tau$. Let's verify this. First, since $\tau$ is the discrete topology we see that every subset of $X$ is contained in $\tau$. For each $B=\{x\}\in {\mathcal {B}}$ we therefore have that:

${\begin{aligned}\quad B=\{x\}\in {\mathcal {B}}\in \tau \end{aligned}}$

For the second condition, let $U\in \tau$. Then since $\tau$ is the discrete topology, we have that $U\subseteq X$. For all $x\in U$, we have that $U$ can be expressed as the union of some collection of sets in ${\mathcal {B}}$. In particular, for each $U\in \tau$ we have that:

${\begin{aligned}\quad U=\bigcup _{x\in U}\{x\}\end{aligned}}$

Therefore ${\mathcal {B}}=\{\{x\}:x\in X\}$ is a base of the discrete topology.

Example 2

For another example, consider the set $X=\{a,b,c,d\}$ and the following topology on $X$:

${\begin{aligned}\quad \tau =\{\emptyset ,\{a\},\{d\},\{a,d\},\{b,c\},\{a,b,c\},\{b,c,d\},X\}\end{aligned}}$

Consider the collection of open sets ${\mathcal {B}}=\{\{a\},\{d\},\{b,c\}\}$. We claim that ${\mathcal {B}}$ is a base of $\tau$. Clearly all of the sets in ${\mathcal {B}}$ are contained in $\tau$, so every set in ${\mathcal {B}}$ is open.

For the second condition, we only need to show that the remaining open sets in $\tau$ that are not in ${\mathcal {B}}$ can be obtained by taking unions of elements in ${\mathcal {B}}$. The $\emptyset \in \tau$ can be obtained by taking the empty union of elements in ${\mathcal {B}}$. Furthermore:

${\begin{aligned}\quad &\{a\}\cup \{d\}=\{a,d\}\\\quad &\{a\}\cup \{b,c\}=\{a,b,c\}\\\quad &\{a\}\cup \{b,c\}\cup \{d\}=X\end{aligned}}$

Therefore every $U\in \tau$ is the union of some collection of sets from ${\mathcal {B}}$, so ${\mathcal {B}}$ is a base of $\tau$.

Example 3

If $\mathbb {R}$ has the usual Euclidean topology, then the collection:

${\begin{aligned}\quad {\mathcal {B}}=\{(a,b):a,b\in \mathbb {R} ,a<b\}\end{aligned}}$

(The collection of bounded open intervals) is a base for the Euclidean topology.

Example 4

If $X$ is any metric space, then the collection:

${\begin{aligned}\quad {\mathcal {B}}=\{B(x,\epsilon ):x\in X,\epsilon >0\}\end{aligned}}$

(The collection of open balls relative to the metric defined on $X$) is a base for the topology resulting from the metric on $X$.

Licensing

Content obtained and/or adapted from:

• Bases of a topology, mathonline.wikidot.com under a CC BY-SA license