We know that a topological space is said to be Noetherian if it satisfy d.c.c. on closed sets, i.e., every descending chain of closed subsets terminates. If in a topological space d.c.c. holds for irreducible closed sets is the topological space Noetherian ? ($Y \subset X$ is said to be irreducible if whenever $Y=Y_1 \cup Y_2$ where $Y_1$ and $Y_2$ are closed in $Y$ either $Y_1=Y$ or $Y_2=Y$)
I am not sure if it is true or not. Help me. Thanks.