Definition:Generated Topology

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Definition

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:

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


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


Definition 1

Define:

$\displaystyle \BB = \set {\bigcap \FF: \FF \subseteq \SS, \FF \text{ is finite} }$

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:

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


Definition 2

The topology generated by $\mathcal S$, denoted $\tau \left({\mathcal S}\right)$, is defined as the unique topology on $X$ that satisfies the following axioms:

$\left({1}\right): \quad \mathcal S \subseteq \tau \left({\mathcal S}\right)$
$\left({2}\right): \quad$ For any topology $\mathcal T$ on $X$, the implication $\mathcal S \subseteq \mathcal T \implies \tau \left({\mathcal S}\right) \subseteq \mathcal T$ holds.

That is, $\tau \left({\mathcal S}\right)$ is the coarsest topology on $X$ for which every element of $\mathcal S$ is open.