# Definition:Sub-Basis

## Definition

### Analytic Sub-Basis

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

Let $\SS \subseteq \tau$.

Define:

$\displaystyle \BB = \set {\bigcap \FF: \FF \subseteq \SS, \FF \text{ is finite} }$

That is, $\BB$ is the set of all finite intersections of sets in $\SS$.

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

Define:

$\displaystyle \tau' = \set {\bigcup \AA: \AA \subseteq \BB}$

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

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

Then $\SS$ 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.