# Definition:Basis (Topology)

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## Definition

### Analytic Basis

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 $\BB$.

### Synthetic Basis

A **synthetic basis on $S$** is a subset $\BB \subseteq \powerset S$ of the power set of $S$ such that:

\((\text B 1)\) | $:$ | $\BB$ is a cover for $S$ | ||||||

\((\text B 2)\) | $:$ | \(\displaystyle \forall U, V \in \BB:\) | $\exists \AA \subseteq \BB: U \cap V = \bigcup \AA$ |

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

## Specification of Topology

As a way of specifying a particular topology on a set, we can say:

In such a case, those sets should all satisfy the axioms for a synthetic basis $(\text B 1)$ and $(\text B 2)$.

## Also known as

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

## 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**.