Definition:Generated Topology

Topology Generated by Synthetic Basis

Let $S$ be a set.

Let $\BB$ be a synthetic basis of $S$.

Definition 1

The topology on $S$ generated by $\BB$ is defined as:

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

That is, the set of all unions of sets from $\BB$.

Definition 2

The topology on $S$ generated by $\BB$ is defined as:

$\tau = \set {U \subseteq S: U = \bigcup \set {B \in \BB: B \subseteq U}}$

Definition 3

The topology on $S$ generated by $\BB$ is defined as:

$\tau = \set {U \subseteq S: \forall x \in U: \exists B \in \BB: x \in B \subseteq U}$

Topology Generated by Synthetic Sub-Basis

Let $X$ be a set.

Let $\SS \subseteq \powerset X$ 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.