Definition:Topology Generated by Synthetic Sub-Basis/Definition 1

Definition

Let $S$ be a set.

Let $\SS \subseteq \powerset S$ be a synthetic sub-basis on $S$.

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.

The topology generated by $\SS$, denoted $\map \tau \SS$, is defined as:

$\ds \map \tau \SS = \set {\bigcup \AA: \AA \subseteq \BB}$

It follows directly from Synthetic Basis formed from Synthetic Sub-Basis and Union from Synthetic Basis is Topology that $\map \tau \SS$ is a topology on $S$.

Also see

• Equivalence of Definitions of Topology Generated by Synthetic Sub-Basis

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: Functions