Definition:Topology Generated by Synthetic Sub-Basis

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Definition

Let $X$ be a set.

Let $\mathcal S \subseteq \mathcal P \left({X}\right)$ be a synthetic sub-basis on $X$.


Definition 1

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:

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


Definition 2

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