# Definition:Sub-Basis

## 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).

## 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.