diff --git a/spaces/S000199/properties/P000039.md b/spaces/S000199/properties/P000039.md
index afd3e05f9..7edff5528 100644
--- a/spaces/S000199/properties/P000039.md
+++ b/spaces/S000199/properties/P000039.md
@@ -4,5 +4,6 @@ property: P000039
 value: true
 ---
 
-Every pair of basic open sets $(\leftarrow,m)$ and $(\leftarrow,n)$
-intersects.
+The topology is a chain under inclusion;
+that is,
+for each pair of open sets, one is contained in the other.
diff --git a/spaces/S000199/properties/P000040.md b/spaces/S000199/properties/P000040.md
index db15ea84b..b25304d62 100644
--- a/spaces/S000199/properties/P000040.md
+++ b/spaces/S000199/properties/P000040.md
@@ -4,4 +4,6 @@ property: P000040
 value: true
 ---
 
-$0$ is in the closure of every non-empty set.
+The collection of closed sets is a chain under inclusion;
+that is,
+for each pair of closed sets, one is contained in the other.
diff --git a/spaces/S000200/properties/P000039.md b/spaces/S000200/properties/P000039.md
index 567175052..ee8b420d9 100644
--- a/spaces/S000200/properties/P000039.md
+++ b/spaces/S000200/properties/P000039.md
@@ -4,5 +4,6 @@ property: P000039
 value: true
 ---
 
-Every pair of basic open sets $[m,\rightarrow)$ and $[n,\rightarrow)$
-intersects.
+The topology is a chain under inclusion;
+that is,
+for each pair of open sets, one is contained in the other.
diff --git a/spaces/S000200/properties/P000040.md b/spaces/S000200/properties/P000040.md
index ba0f28b83..644271c6f 100644
--- a/spaces/S000200/properties/P000040.md
+++ b/spaces/S000200/properties/P000040.md
@@ -4,6 +4,6 @@ property: P000040
 value: true
 ---
 
-Let $m\in M$ and $n\in N$ be members of closed sets $M,N$.
-Then every neighborhood of $m$ contains $n$, so $m$ is a
-limit point of $N$, showing $m\in M\cap N$.
+The collection of closed sets is a chain under inclusion;
+that is,
+for each pair of closed sets, one is contained in the other.