Definition:Sub-Basis/Synthetic Sub-Basis

Definition

Let $S$ be a set.

A synthetic sub-basis on $S$ is any subset $\SS \subseteq \powerset S$ of the power set of $S$.

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)

• Synthetic Basis formed from Synthetic Sub-Basis

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