Definition:Filter Sub-Basis
Jump to navigation
Jump to search
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