# Definition:Sub-Basis

## Contents

## Definition

### Analytic Sub-Basis

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

Let $\mathcal S \subseteq \tau$.

Define:

- $\displaystyle \mathcal B = \left\{{\bigcap \mathcal F: \mathcal F \subseteq \mathcal S, \, \mathcal F \text{ is finite}}\right\}$

That is, $\mathcal B$ is the set of all finite intersections of sets in $\mathcal S$.

Note that $\mathcal F$ is allowed to be empty in the above definition.

Define:

- $\displaystyle \tau' = \left\{{\bigcup \mathcal A: \mathcal A \subseteq \mathcal B}\right\}$

Suppose that $\tau \subseteq \tau'$.

That is, suppose that every $U \in \tau$ is a union of finite intersections of sets in $\mathcal S$, together with $\varnothing$ and $S$ itself.

Then $\mathcal S$ is called an **analytic sub-basis** for $\tau$.

### Synthetic Sub-Basis

Let $X$ be a set.

A **synthetic sub-basis on $X$** is *any* subset $\mathcal S \subseteq \mathcal P \left({X}\right)$ of the power set of $X$.

## Also known as

Some sources do not hyphenate **sub-basis** but instead render it as **subbasis**.

Some sources use the term **sub-base** (or **subbase**).

## Also see

- Definition:Basis (Topology)
- Definition:Filter Sub-Basis
- Definition:Topology Generated by Synthetic Sub-Basis

## Linguistic Note

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

This is properly pronounced **sub-bay-seez**, rather than **sub-bay-siz**, deriving as it does from the Greek plural form of nouns ending **-is**.