# Definition:Basis (Topology)/Analytic Basis

< Definition:Basis (Topology)(Redirected from Definition:Analytic Basis)

Jump to navigation
Jump to search
## Definition

### Definition 1

Let $\struct {S, \tau}$ be a topological space.

An **analytic basis for $\tau$** is a subset $\BB \subseteq \tau$ such that:

- $\ds \forall U \in \tau: \exists \AA \subseteq \BB: U = \bigcup \AA$

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

### Definition 2

Let $\struct {S, \tau}$ be a topological space.

Let $\BB \subseteq \tau$.

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

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

## 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**.