# Definition:Sub-Basis/Synthetic Sub-Basis

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## Definition

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 distinguish between a **synthetic sub-basis** and an **analytic sub-basis**, and instead use the definition for an **analytic sub-basis** and just call that 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

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

## 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): $3.3$: Sub-bases and weak topologies