Title: Noetherian topological spaces

Abstract: A topological space is noetherian if it satisfies the  descending chain condition for closed sets. 

The topological spaces met in algebraic geometry are  usually noetherian.

I will prove some countability properties for collections of closed sets and give applications to commutative algebra.

E.g.  I will prove that every totally ordered set of closed sets in a noetherian space  (with some additional condition) must be countable.