# Chain of interior of closed set

It is well known that a topological space with asending chain condition for open subsets is called Noetherian. Is there any characterizations or a nice property for a Hausdorff topological space such that every asending chain of interior of closed subsets stops, that is, every chain of the form $int (C_1)\subseteq int (C_2)\subseteq...$ stops, where $C_1$, $C_2$,... are closed?

• Yes, there is. The space must be finite. – Ramiro de la Vega Feb 25 '17 at 3:12
• @RamirodelaVega Is it obvious that mere Hausdorffness (rather than regularity) suffices? – მამუკა ჯიბლაძე Feb 26 '17 at 9:00
• Perhaps it is not so obvious, I posted some details. – Ramiro de la Vega Feb 26 '17 at 11:39

If $X$ is an infinite Hausdorff space, we can inductively construct a sequence of points $\{x_n\}$ and a sequence of pairwise disjoint open sets $\{U_n\}$ with $x_n \in U_n$ for each $n \in \mathbb{N}$. Now just let $C_n$ be the closure of the union of the $U_m$ with $m \leq n$. Note that $int(C_n) \subseteq int(C_{n+1})$ and $x_{n+1} \in int(C_{n+1}) \setminus C_n$ for every $n$.
If $X$ is a finite Hausdorff space, any chain of sets must stop.