Base

Definition (Base/Basis)

$\beta \subseteq \mathcal{P}(X)$ is a base (or basis) for the topology $\tau$ if any element $U\in \tau$ can be written as a union of elements in $\beta$.

Examples

In $\mathbb{R}$ with the standard topology, the set of all open intervals $(a,b)$ is a basis for the topology, since any open set of $\mathbb{R}$ can be written as a union of open intervals.

In a Metric Space $(X,d)$, the $\{B_{\epsilon }(x),\epsilon >0,x\in X\}$ forms a basis for the associated topology.

Theorem (Minimum Basis for Topology)

What conditions on $\beta$ are necessary for it to generate a topology?

Suppose $\beta \subseteq \mathcal{P}(X)$ satisfies

1. $\forall x\in X,\exists B\in \beta$ such that $x\in B$.
   1. All we are really saying is that $X$ can be written as a union of elements of $\beta$.
2. $\forall B_1,B_2\in \beta$ and any element $x\in B_1\cap B_2,\exists B_3\in \beta$ such that $x\in B_3\subseteq B_1\cap B_2$.
   1. Also equivalent, is $\forall B_1,B_2\in \beta ,B_1\cap B_2$ can be written as a union of elements of $\beta$.
   2. However $2/⟹2.1$ because we can write $B_1\cap B_2={\bigcup }_{x\in B_1\cap B_2}{B}_3$

then $\beta$ is a basis for a for a topology on $\tau$. The converse is also true.

Proof: $(⟸)$ Certainly if $\beta$ is a base for $\tau$, then the two conditions are satisfied since $X$ can be

1. written as a union of elements of $\beta$.
2. If $B_1,B_2\in \beta$ then $B_1\cap B_2$ is open. Therefore it can be written as a union by $2.1$.

$(⟹)$ Merely check the finite intersection axiom. It is enough to do it pairwise and check

$$\left(\bigcup _{i\in I}{B}_i\right)\cap \left(\bigcup _{j\in J}{B}_j\right)=\bigcup _{i,j}{B}_i\cap B_j$$

for $B_i,B_j\in \beta$. Then by $2.1$ this is just the union of elements of $\beta$.

Example 1

In a Metric Space $X$, the set of all

$$B_{\epsilon }(x)\forall x\in X,\forall \epsilon >0$$

forms a base.

Example 2

We can define a topology on $\mathbb{R}$ by setting

$$\beta =\{\text{all intervals of the form}[a,b):a<b\}$$

We check: intersecting any two such things gives another one, so this is a basis.

Example 3

We can define topologies on spaces that are not metric spaces.

The real line without a point. $\mathbb{R}\cap \{0^{\prime }\}$ where

$$\beta =\{\text{all intersections of the form }(a,b)\text{ and where 0’ replaces 0, if 0 }\in (a,b)\}$$

Clearly any intersection of any two such basic open intervals is either another element of the same form or $(-a,0)\cup (0,b)$ is the union of base elements.

Remark

Bases are usually not unique. Any representation of an open set $U\in T$ as a union of base elements $U={\bigcup }_{i\in I}{B}_i$ is almost certainly not unique.

Definition (Product of Spaces Base)

See Definition (Product of Spaces Base).