Definition:Minimal Topology

Definition

Let $S$ be a set and let $\mathcal P \left({S}\right)$ be the power set of $S$.

Let $\Theta_S$ be the set of all topologies on $S$:

$\Theta_S = \left\{{\tau \in \mathcal P \left({S}\right): \tau}\right.$ is a topology on $\left.{S}\right\}$

Let $\Phi: \Theta_S \to \left\{{T, F}\right\}$ be a propositional function on $\Theta_S$.

Let $\vartheta \in \Theta_S$ have the property that $\Phi \left({\vartheta}\right)$ and:

$\forall \tau \in \Theta_S: \Phi \left({\tau}\right) \implies \vartheta \subseteq \tau$

That is, $\vartheta$ is the coarsest topology on $S$ which satisfies the propositional function $\Phi$.

Then $\vartheta$ is the minimal topology satisfying $\Phi$.