In the proof of noetherian space implies quasicompact I am reading, it goes as follows: Let $X$ be a noetherian space.
Let $U$ be the collection of open subsets of $X$ that can be expressed as a finite union of opens sets in the covering. If $U$ does not contain $X$, then there exists an infinite ascending chain of sets in $U$ (axiom of dependent choice), hence contradiction.
I know it is similar to Why is axiom of choice needed? (Equivalent conditions for Noetherian) , but I am just not seeing why the axiom of dependent choice is necessary here... I would greatly appreciate some explanation. Thank you!
Ps by Noethrian space, I mean Noetheiran topological space: Every descending chain of closed subsets of $X$ eventually becomes constant.