# Definition:Basis (Topology)/Analytic Basis

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## Contents

## Definition

### Definition 1

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 $\mathcal B$.

### Definition 2

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

Let $\mathcal B \subseteq \tau$.

Then $\mathcal B$ is an analytic basis for $\tau$ if and only if:

- $\forall U \in \tau: \forall x \in U: \exists V \in \mathcal B: x \in V \subseteq U$

## Also known as

Some sources do not distinguish between an **analytic basis** and a synthetic basis, and instead use this definition and call it a **basis**.

## Also see

- Results about
**bases**can be found here.

## Linguistic Note

The plural of **basis** is **bases**.

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