Definition:Minimal Topology

Definition

Let $S$ be a set.

Let $\powerset S$ be the power set of $S$.

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

$\Theta_S = \leftset {\tau \in \powerset S: \tau}$ is a topology on $\rightset S$

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

Let $\vartheta \in \Theta_S$ have the property that $\map \Phi \vartheta$ and:

$\forall \tau \in \Theta_S: \map \Phi \tau \implies \vartheta \subseteq \tau$

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

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