# Definition:Basis (Topology)

## Contents

## 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.