Equivalence of Definitions of Topology Generated by Synthetic Basis

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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:

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

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 that $U \in \tau$. Then, by definition:

$\exists \AA \subseteq \BB: U = \bigcup \AA$

By Union is Smallest Superset: General Result, we have:

$\forall B \in \AA: B \subseteq U$

By the definition of a 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:

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

By Union is Smallest Superset: General Result:

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

By definition of set equality:

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

$\Box$


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:

$U \subseteq \bigcup \left\{{B \in \mathcal B: B \subseteq U}\right\}$

By Union is Smallest Superset: General Result:

$\bigcup \left\{{B \in \mathcal B: B \subseteq U}\right\} \subseteq U$


By definition of set equality:

$U = \bigcup \left\{{B \in \mathcal B: B \subseteq U}\right\}$

The result follows.

$\blacksquare$