User:Dfeuer/topext

Let $X$ be a set.

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

Let $\SS$ be a set of subsets of $X$.

Let $\mathscr S = \set {Q \in \powerset {\powerset X} \mid \SS \subseteq Q}$.

Let $f: \mathscr S \to \mathscr S$ be defined by

$\ds \map f Q = \bigcup_{p \mathop \in \CC} p^{-1} \sqbrk Q$.

That is, let $\map f Q$ be the set of preimages of elements of $Q$ under elements of $\CC$.

Let $g: \mathscr S \to \mathscr S$ with $\map g Q$ 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, $\MM$.

If I'm not mistaken, $\MM$ is then the coarsest topology on $X$ in which each element of $\SS$ is open and each element of $\CC$ is continuous.