# Definition:Topology Generated by Synthetic Basis

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## Definition

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:

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

## Also see

- Union from Synthetic Basis is Topology, which proves that $\tau$ is a topology on $S$