Definition:Filter Sub-Basis

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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.