pi-base · prabau · Aug 13, 2026 · Aug 19, 2026
diff --git a/spaces/S000125/README.md b/spaces/S000125/README.md
@@ -10,11 +10,14 @@ refs:
 - wikipedia: Knaster-Kuratowski_fan
   name: Knaster-Kuratowski fan
 ---
-For any $a \in [0,1]$, let $L(a)$ be the line segment from $(a,0)$ to $p = (\frac{1}{2}, \frac{1}{2})$. Let $\mathcal{C}$ be the middle-thirds Cantor set in the unit interval, $\mathcal{E}$ the endpoints of the removed intervals and $\mathcal{F} = \mathcal{C} \setminus \mathcal{E}$. Define $A = \{(x,y) \in L(c)\ |\ c \in \mathcal{E}, y \in \mathbb{Q}\}$ and $B = \{(x,y) \in L(c)\ |\ c \in \mathcal{F}, y \not\in \mathbb{Q}\}$. This space is $X = A \cup B \subset \mathbb{R}^2$ with the subspace topology.
+
+$X$ is a subspace of {S176} defined as follows.
+For any $a \in [0,1]$, let $L(a)$ be the closed line segment from $(a,0)$ to $p = (\frac{1}{2}, \frac{1}{2})$.
+Let $C$ be the middle-thirds Cantor set in the unit interval, $E$ the set of endpoints of the removed intervals and $F = C\setminus E$.
+Define $A = \{(x,y) \in L(c):c \in E, y \in \mathbb Q\}$ and $B = \{(x,y) \in L(c):c \in F, y \not\in \mathbb Q\}$.
+Then take $X = A \cup B\subseteq\mathbb R^2$ with the subspace topology.
 
 The subspace $X\setminus\{p\}$ obtained by removing the apex point is {S126}.
 
 Defined as counterexample #128 ("Cantor's Leaky Tent")
 in {{zb:0386.54001}}.
-
-<!-- ![](http://i.imgur.com/P36Jx0z.png?1) -->
diff --git a/spaces/S000125/properties/P000022.md b/spaces/S000125/properties/P000022.md
diff --git a/spaces/S000125/properties/P000023.md b/spaces/S000125/properties/P000023.md
diff --git a/spaces/S000125/properties/P000041.md b/spaces/S000125/properties/P000041.md
@@ -2,10 +2,7 @@
 space: S000125
 property: P000041
 value: false
-refs:
-- zb: "0386.54001"
-  name: Counterexamples in Topology
 ---
 
-Asserted in the General Reference Chart for space #128 in
-{{zb:0386.54001}}.
+{S126} is an open subset of $X$,
+and {S126|P41}.
diff --git a/spaces/S000125/properties/P000055.md b/spaces/S000125/properties/P000055.md
@@ -0,0 +1,9 @@
+---
+space: S000125
+property: P000055
+value: false
+---
+
+Take some $c\in E$.
+The closed set $L(c)\cap X\subseteq X$ contains a closed set homeomorphic to {S27}.
+And {S27|P55}.
diff --git a/spaces/S000125/properties/P000066.md b/spaces/S000125/properties/P000066.md
@@ -0,0 +1,9 @@
+---
+space: S000125
+property: P000066
+value: false
+---
+
+Take some $c\in F$.
+The closed set $L(c)\cap X\subseteq X$ contains a closed set homeomorphic to {S28}.
+And {S28|P66}.