# Definition:Basis (Topology)

## Definition

### Analytic Basis

Let $\left({S, \tau}\right)$ be a topological space.

An analytic basis for $\tau$ is a subset $\mathcal B \subseteq \tau$ such that:

$\displaystyle \forall U \in \tau: \exists \mathcal A \subseteq \mathcal B: U = \bigcup \mathcal A$

That is, such that for all $U \in \tau$, $U$ is a union of sets from $\mathcal B$.

### Synthetic Basis

Let $S$ be a set.

#### Definition 1

A synthetic basis on $S$ is a subset $\mathcal B \subseteq \mathcal P \left({S}\right)$ of the power set of $S$ such that:

 $(B1)$ $:$ $\mathcal B$ is a cover for $S$ $(B2)$ $:$ $\displaystyle \forall U, V \in \mathcal B:$ $\exists \mathcal A \subseteq \mathcal B: U \cap V = \bigcup \mathcal A$

That is, the intersection of any pair of elements of $\mathcal B$ is a union of sets of $\mathcal B$.

#### Definition 2

A synthetic basis on $S$ is a subset $\mathcal B \subseteq \mathcal P \left({S}\right)$ of the power set of $S$ such that:

$\mathcal B$ is a cover for $S$
$\forall U, V \in \mathcal B: \forall x \in U \cap V: \exists W \in \mathcal B: x \in W \subseteq U \cap V$

## Countable Basis

A countable (analytic) basis for a topology $\tau$ is an analytic basis for $\tau$ which is a countable set.

## Also known as

A basis can also be seen referred to as a base.

## Linguistic Note

The plural of basis is bases.

This is properly pronounced bay-seez, not bay-siz.

## Also see

• Results about bases can be found here.