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?



Name for $\omega_1$-DCC / Noetherian condition?
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.


