pi-base · felixpernegger · Mar 30, 2026 · Mar 30, 2026 · May 15, 2026 · May 15, 2026
diff --git a/spaces/S000136/properties/P000019.md b/spaces/S000136/properties/P000019.md
@@ -2,9 +2,6 @@
 space: S000136
 property: P000019
 value: false
-refs:
-- zb: "0386.54001"
-  name: Counterexamples in Topology
 ---
 
 Closed subspaces of {P19} spaces are {P19}, but the closed subspace {S137} of this space is not {P19}.
diff --git a/spaces/S000136/properties/P000051.md b/spaces/S000136/properties/P000051.md
@@ -0,0 +1,9 @@
+---
+space: S000136
+property: P000051
+value: true
+---
+
+For $r \in \mathbb{R}$ consider the characteristic function $\chi_{\{r\}}:2^\mathbb{R}\to 2$. Then $\chi_{\{r\}} \cap M = x_r$, therefore $M$ is discrete as a subspace.
+
+Thus if $A \subseteq X$ contains no elements of $F$ we are done, otherwise $A$ already contains an isolated point by definition.
diff --git a/spaces/S000136/properties/P000139.md b/spaces/S000136/properties/P000139.md
diff --git a/spaces/S000137/properties/P000051.md b/spaces/S000137/properties/P000051.md
@@ -0,0 +1,7 @@
+---
+space: S000137
+property: P000051
+value: true
+---
+
+$X$ is a subspace of {S136} and {S136|P51}.