One calls a topological space noetherian if every decreasing chain of closed subsets becomes stationary.
In a lecture it was remarked that this is equivalent to every family $M$ of closed sets having minimal elements. However to me it looks like the direction
$X$ noetherian $\implies$ every family of closed sets $M$ has minimal element
requires use of Zorns Lemma: Every decreasing chain in $M$ must become stationary and thus be bounded, from Zorns Lemma it follows that a minimal element exists.
In the exercises we also showed that noetherian $\iff$ every open set is quasi-compact. Also here my proof of the $\implies$ direction used the Axiom of Choice.
My question: Are these statements actually dependent of axiom of choice or are my proofs just bad?