# Definition:Filter Sub-Basis

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## Definition

Let $S$ be a set.

Let $\powerset S$ denote the power set of $S$.

Let $\BB \subset \powerset S$ be a set of subsets of $\powerset S$ which satisfies the finite intersection property.

That is, the intersection of any finite number of sets in $\BB$ is not empty.

Then $\BB$, together with the finite intersections of all its elements, is a basis for a filter $\FF$ on $S$.

Thus $\BB$ is a **sub-basis** for $\FF$.

## 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

## 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

- 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: Filters