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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?

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

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