Equivalence of Definitions of Topology Generated by Synthetic Basis
Theorem
Let $S$ be a set.
Let $\BB$ be a synthetic basis on $S$.
The following definitions of the concept of Topology Generated by Synthetic Basis are equivalent:
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:
- $\tau = \set {U \subseteq S: \forall x \in U: \exists B \in \BB: x \in B \subseteq U}$
Proof
Definition 1 iff Definition 2
Trivially, the reverse implication holds, as $\set {B \in \BB: B \subseteq U} \subseteq \BB$.
We now show that the forward implication holds.
Suppose $U \in \tau$.
Then, by definition:
- $\ds \exists \AA \subseteq \BB: U = \bigcup \AA$
By Union is Smallest Superset: General Result:
- $\forall B \in \AA: B \subseteq U$
By definition of subset, it follows that:
- $\AA \subseteq \set {B \in \BB: B \subseteq U}$
From Union of Subset of Family is Subset of Union of Family:
- $\ds U = \bigcup \AA \subseteq \bigcup \set {B \in \BB: B \subseteq U}$
By Union is Smallest Superset: General Result:
- $\ds \bigcup \set {B \in \BB: B \subseteq U} \subseteq U$
By definition of set equality:
- $\ds U = \bigcup \set {B \in \BB: B \subseteq U}$
$\Box$
Definition 1 iff Definition 3
From Set is Subset of Union: General Result, the forward implication directly follows.
We now show that the reverse implication holds.
By hypothesis, we have that:
- $\ds U \subseteq \bigcup \set {B \in \BB: B \subseteq U}$
By Union is Smallest Superset: General Result:
- $\ds \bigcup \set {B \in \BB: B \subseteq U} \subseteq U$
By definition of set equality:
- $\ds U = \bigcup \set {B \in \BB: B \subseteq U}$
The result follows.
$\blacksquare$