# Definition:Sub-Basis/Analytic Sub-Basis

## Definition

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

Let $\SS \subseteq \tau$.

Define:

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

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:

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

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 $\tau$.

## Also known as

Some sources do not distinguish between an **analytic sub-basis** and a **synthetic sub-basis**, and instead use this definition and call it a **sub-basis**.

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

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

## Also see

- Synthetic Sub-Basis and Analytic Sub-Basis are Compatible
- Definition:Synthetic Sub-Basis
- Definition:Basis (Topology)

- Results about
**Sub-Bases**can be found**here**.

## 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**.

## Sources

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.3$: Sub-bases and weak topologies - 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction