Nested interval topology S44 is quasi-sober P192

properties/P000192.md

@@ -6,8 +6,8 @@ refs:
   name: Sober spaces and sober sets (Echi & Lazaar)
 ---
 
-Every nonempty irreducible closed subset of $X$ is the closure of a point of $X$, not necessarily unique; that is, every nonempty irreducible closed subset has at least one *generic point*.
-Here, a subset of $X$ is called *irreducible* if it is {P39} with the subspace topology.
+Every non-{P137} {P39} closed subspace of $X$ is
+{P201}.
 
 Equivalently, the Kolmogorov quotient of the space is {P73}. (See {T512}.)
 

spaces/S000044/properties/P000192.md

@@ -0,0 +1,10 @@
+---
+space: S000044
+property: P000192
+value: true
+---
+
+The non-empty closed subsets of {S44} are the closed intervals 
+$[\frac{n-1}{n}, 1)$, and each is {P201} witnessed by $\frac{n-1}{n}$: the
+nonempty open proper subsets of $[\frac{n-1}{n}, 1)$ are $[\frac{n-1}{n}, \frac{N-1}{N})$ for $N > n$,
+and each contains $\frac{n-1}{n}$.