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