diff --git a/properties/P000192.md b/properties/P000192.md index 19921e9af9..b8586af8b7 100644 --- a/properties/P000192.md +++ b/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}.) diff --git a/spaces/S000044/properties/P000086.md b/spaces/S000044/properties/P000086.md deleted file mode 100644 index 2c23509426..0000000000 --- a/spaces/S000044/properties/P000086.md +++ /dev/null @@ -1,8 +0,0 @@ ---- -space: S000044 -property: P000086 -value: false ---- - -$\frac{2}{5}$ belongs to a single closed set $(0,1)$, but -$\frac{3}{5}$ belongs to two closed sets $(0,1)$ and $[1/2,1)$. diff --git a/spaces/S000044/properties/P000192.md b/spaces/S000044/properties/P000192.md new file mode 100644 index 0000000000..f5806e2194 --- /dev/null +++ b/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 $\frac{n-1}{n}$.