TITLE: Name for $\omega_1$-DCC / Noetherian condition?
QUESTION [6 upvotes]: I recently asked (and then answered) this question: 
https://math.stackexchange.com/questions/2756777/decreasing-sequence-of-closed-sets-in-a-separable-metric-space.
In a separable metric space there is no strictly decreasing sequence of closed sets $(X_\alpha)_{\alpha<\omega_1}$, where $X_\beta\supsetneq X_\alpha$ iff $\beta<\alpha$.
This sounds a lot like the Descending Chain Condition in Algebra.  I assume it comes up frequently in topology (maybe not?).  In that case, is there a name for it?

REPLY [7 votes]: For any topological space $X$, the following statements are easily seen to be equivalent:
(1) there is no strictly decreasing $\omega_1$-sequence of closed sets in $X$;
(2) there is no strictly increasing $\omega_1$-sequence of open sets in $X$;
(3) every open subspace of $X$ is Lindelöf;
(4) every subspace of $X$ is Lindelöf.
Spaces satisfying those equivalent conditions are called hereditarily Lindelöf spaces by some, strongly Lindelöf spaces by others.