\begin{definition}[Definition:Minimal Topology]
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 $\Phi$'''.
Category:Definitions/Topology
\end{definition}