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 $S$