User:Dfeuer/topext

Let $X$ be a set.

Let $\mathcal C$ be a set of functions from $X$ to $X$ which contains the identity function.

Let $\mathcal S$ be a set of subsets of $X$.

Let $\mathscr S = \{ Q \in \mathcal P(\mathcal P(X)) \mid \mathcal S \subseteq Q \}$.

Let $f: \mathscr S \to \mathscr S$ be defined by
 * $\displaystyle f(Q) = \bigcup_{p\in\mathcal C}p^{-1}[Q]$.

That is, let $f(Q)$ be the set of preimages of elements of $Q$ under elements of $\mathcal C$.

Let $g: \mathscr S \to \mathscr S$ with $g(Q)$ being the topology on $X$ generated by the subbasis $Q$.

Let $h = g \circ f$.

Then $h$ is an increasing function on $\mathscr S$.

I believe that $\mathscr S$ is a complete lattice under inclusion.

Thus by the Knaster-Tarski Lemma, $h$ has a least fixed point, $\mathcal M$.

If I'm not mistaken, $\mathcal M$ is then the coarsest topology on $X$ in which each element of $\mathcal S$ is open and each element of $\mathcal C$ is continuous.