Definition:Topology Generated by Synthetic Sub-Basis/Definition 2

Definition

Let $X$ be a set.

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

The topology generated by $\SS$, denoted $\map \tau \SS$, is defined as the unique topology on $X$ that satisfies the following axioms:

$(1): \quad \SS \subseteq \map \tau \SS$

$(2): \quad$ For any topology $\TT$ on $X$, the implication $\SS \subseteq \TT \implies \map \tau \SS \subseteq \TT$ holds.

That is, $\map \tau \SS$ is the coarsest topology on $X$ for which every element of $\SS$ is open.

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