From bd051f85ece898083235e54186087ff979cf2d19 Mon Sep 17 00:00:00 2001
From: Patrick Rabau <70125716+prabau@users.noreply.github.com>
Date: Tue, 3 Feb 2026 18:50:18 -0500
Subject: [PATCH 1/6] alpha_i preserved by finite products

---
 properties/P000210.md                | 3 +++
 properties/P000211.md                | 3 +++
 properties/P000212.md                | 3 +++
 properties/P000213.md                | 3 +++
 properties/P000214.md                | 3 +++
 spaces/S000078/README.md             | 3 ++-
 spaces/S000078/properties/P000081.md | 3 ++-
 spaces/S000078/properties/P000210.md | 8 ++++++++
 spaces/S000091/properties/P000210.md | 9 +++++++++
 spaces/S000191/properties/P000210.md | 8 ++++++++
 spaces/S000218/properties/P000210.md | 8 ++++++++
 11 files changed, 52 insertions(+), 2 deletions(-)
 create mode 100644 spaces/S000078/properties/P000210.md
 create mode 100644 spaces/S000091/properties/P000210.md
 create mode 100644 spaces/S000191/properties/P000210.md
 create mode 100644 spaces/S000218/properties/P000210.md

diff --git a/properties/P000210.md b/properties/P000210.md
index 7ae5540a4a..305f30f4fb 100644
--- a/properties/P000210.md
+++ b/properties/P000210.md
@@ -10,6 +10,8 @@ refs:
     name: Subsets of ${}^\omega\omega$ and the Fréchet-Urysohn and $\alpha_i$-properties (P. Nyikos)
   - zb: "0275.54004"
     name: The frequency spectrum of a topological space and the classification of spaces (A. V. Arkhangel'skii)
+  - mathse: 5122756
+    name: Are the Arkhangel'skii $\alpha_i$ properties preserved by finite products?
 ---
 
 For every $x\in X$ and every sequence of countably infinite sets $S_1,S_2,\dots\subseteq X$ with
@@ -35,3 +37,4 @@ are due to Arkhangel'skii ({{zb:0275.54004}}).
 #### Meta-properties
 
 - This property is hereditary.
+- This property is preserved by finite products (see {{mathse:5122756}}).
diff --git a/properties/P000211.md b/properties/P000211.md
index 45bdf65186..f358bce0b8 100644
--- a/properties/P000211.md
+++ b/properties/P000211.md
@@ -8,6 +8,8 @@ refs:
     name: Arhangel'skii sheaf amalgamations in topological groups (Tsaban & Zdomskyy)
   - zb: "0774.54019"
     name: Subsets of ${}^\omega\omega$ and the Fréchet-Urysohn and $\alpha_i$-properties (P. Nyikos)
+  - mathse: 5122756
+    name: Are the Arkhangel'skii $\alpha_i$ properties preserved by finite products?
 ---
 
 For every $x\in X$ and every sequence of pairwise disjoint countably infinite sets
@@ -34,3 +36,4 @@ This property was introduced by Nyikos in {{zb:0774.54019}}.
 #### Meta-properties
 
 - This property is hereditary.
+- This property is preserved by finite products (see {{mathse:5122756}}).
diff --git a/properties/P000212.md b/properties/P000212.md
index 0489e2f72f..5a77e7cc44 100644
--- a/properties/P000212.md
+++ b/properties/P000212.md
@@ -10,6 +10,8 @@ refs:
     name: Subsets of ${}^\omega\omega$ and the Fréchet-Urysohn and $\alpha_i$-properties (P. Nyikos)
   - zb: "0275.54004"
     name: The frequency spectrum of a topological space and the classification of spaces (A. V. Arkhangel'skii)
+  - mathse: 5122756
+    name: Are the Arkhangel'skii $\alpha_i$ properties preserved by finite products?
 ---
 
 For every $x\in X$ and every sequence of countably infinite sets $S_1,S_2,\dots\subseteq X$ with
@@ -34,3 +36,4 @@ are due to Arkhangel'skii ({{zb:0275.54004}}).
 #### Meta-properties
 
 - This property is hereditary.
+- This property is preserved by finite products (see {{mathse:5122756}}).
diff --git a/properties/P000213.md b/properties/P000213.md
index 8e5da280e1..6bce164dcf 100644
--- a/properties/P000213.md
+++ b/properties/P000213.md
@@ -10,6 +10,8 @@ refs:
     name: Subsets of ${}^\omega\omega$ and the Fréchet-Urysohn and $\alpha_i$-properties (P. Nyikos)
   - zb: "0275.54004"
     name: The frequency spectrum of a topological space and the classification of spaces (A. V. Arkhangel'skii)
+  - mathse: 5122756
+    name: Are the Arkhangel'skii $\alpha_i$ properties preserved by finite products?
 ---
 
 For every $x\in X$ and every sequence of countably infinite sets $S_1,S_2,\dots\subseteq X$ with
@@ -34,3 +36,4 @@ are due to Arkhangel'skii ({{zb:0275.54004}}).
 #### Meta-properties
 
 - This property is hereditary.
+- This property is preserved by finite products (see {{mathse:5122756}}).
diff --git a/properties/P000214.md b/properties/P000214.md
index 47a4b1e41b..a529833ba2 100644
--- a/properties/P000214.md
+++ b/properties/P000214.md
@@ -10,6 +10,8 @@ refs:
     name: Subsets of ${}^\omega\omega$ and the Fréchet-Urysohn and $\alpha_i$-properties (P. Nyikos)
   - zb: "0275.54004"
     name: The frequency spectrum of a topological space and the classification of spaces (A. V. Arkhangel'skii)
+  - mathse: 5122756
+    name: Are the Arkhangel'skii $\alpha_i$ properties preserved by finite products?
 ---
 
 For every $x\in X$ and every sequence of countably infinite sets $S_1,S_2,\dots\subseteq X$ with
@@ -34,3 +36,4 @@ are due to Arkhangel'skii ({{zb:0275.54004}}).
 #### Meta-properties
 
 - This property is hereditary.
+- This property is preserved by finite products (see {{mathse:5122756}}).
diff --git a/spaces/S000078/README.md b/spaces/S000078/README.md
index c63c3adda2..eba30e4fc6 100644
--- a/spaces/S000078/README.md
+++ b/spaces/S000078/README.md
@@ -9,7 +9,8 @@ refs:
     name: Tychonoff plank on Wikipedia
 ---
 
-The space $[0,\omega_1] \times [0,\omega]$ where each factor has the order topology.
+The space $[0,\omega_1] \times [0,\omega]$ where each factor has the order topology;
+that is, the product of {S36} and {S20}.
 
 Defined as counterexample #86 ("Tychonoff Plank")
 in {{zb:0386.54001}}.
diff --git a/spaces/S000078/properties/P000081.md b/spaces/S000078/properties/P000081.md
index 8b6847144e..8b002b025f 100644
--- a/spaces/S000078/properties/P000081.md
+++ b/spaces/S000078/properties/P000081.md
@@ -7,4 +7,5 @@ refs:
   name: Counterexamples in Topology
 ---
 
-{P81} is a hereditary property.  The space contains as a subspace a copy of {S36}, which is not {P81}.
+{S36} is homeomorphic to a subspace of $X$
+and {S36|P81}.
diff --git a/spaces/S000078/properties/P000210.md b/spaces/S000078/properties/P000210.md
new file mode 100644
index 0000000000..cdc82fba96
--- /dev/null
+++ b/spaces/S000078/properties/P000210.md
@@ -0,0 +1,8 @@
+---
+space: S000078
+property: P000210
+value: true
+---
+
+{S36|P210} and {S20|P210},
+hence so is their product.
diff --git a/spaces/S000091/properties/P000210.md b/spaces/S000091/properties/P000210.md
new file mode 100644
index 0000000000..545a3b93ce
--- /dev/null
+++ b/spaces/S000091/properties/P000210.md
@@ -0,0 +1,9 @@
+---
+space: S000091
+property: P000210
+value: true
+---
+
+{S154|P210} and
+{S20|P210},
+hence so is their product, which contains $X$ as a homeomorphic subspace.
diff --git a/spaces/S000191/properties/P000210.md b/spaces/S000191/properties/P000210.md
new file mode 100644
index 0000000000..ec450685bf
--- /dev/null
+++ b/spaces/S000191/properties/P000210.md
@@ -0,0 +1,8 @@
+---
+space: S000191
+property: P000210
+value: true
+---
+
+{S155|P210} and {S20|P210},
+hence so is their product.
diff --git a/spaces/S000218/properties/P000210.md b/spaces/S000218/properties/P000210.md
new file mode 100644
index 0000000000..e8c9f3b448
--- /dev/null
+++ b/spaces/S000218/properties/P000210.md
@@ -0,0 +1,8 @@
+---
+space: S000218
+property: P000210
+value: true
+---
+
+{S35|P210} and {S36|P210},
+hence so is their product.

From b98191eaf92bb2e498e2d1823516747e99b86426 Mon Sep 17 00:00:00 2001
From: Patrick Rabau <70125716+prabau@users.noreply.github.com>
Date: Wed, 4 Feb 2026 01:16:08 -0500
Subject: [PATCH 2/6] alpha_4 not preserved by products

---
 properties/P000214.md | 3 ---
 1 file changed, 3 deletions(-)

diff --git a/properties/P000214.md b/properties/P000214.md
index a529833ba2..47a4b1e41b 100644
--- a/properties/P000214.md
+++ b/properties/P000214.md
@@ -10,8 +10,6 @@ refs:
     name: Subsets of ${}^\omega\omega$ and the Fréchet-Urysohn and $\alpha_i$-properties (P. Nyikos)
   - zb: "0275.54004"
     name: The frequency spectrum of a topological space and the classification of spaces (A. V. Arkhangel'skii)
-  - mathse: 5122756
-    name: Are the Arkhangel'skii $\alpha_i$ properties preserved by finite products?
 ---
 
 For every $x\in X$ and every sequence of countably infinite sets $S_1,S_2,\dots\subseteq X$ with
@@ -36,4 +34,3 @@ are due to Arkhangel'skii ({{zb:0275.54004}}).
 #### Meta-properties
 
 - This property is hereditary.
-- This property is preserved by finite products (see {{mathse:5122756}}).

From f5c87213c7820423ee5804ef27a8d937b37571e8 Mon Sep 17 00:00:00 2001
From: Patrick Rabau <70125716+prabau@users.noreply.github.com>
Date: Mon, 9 Feb 2026 02:33:46 -0500
Subject: [PATCH 3/6] preserved by disjoint unions

---
 properties/P000210.md | 1 +
 properties/P000211.md | 1 +
 properties/P000212.md | 1 +
 properties/P000213.md | 1 +
 properties/P000214.md | 1 +
 5 files changed, 5 insertions(+)

diff --git a/properties/P000210.md b/properties/P000210.md
index 305f30f4fb..a371af8414 100644
--- a/properties/P000210.md
+++ b/properties/P000210.md
@@ -38,3 +38,4 @@ are due to Arkhangel'skii ({{zb:0275.54004}}).
 
 - This property is hereditary.
 - This property is preserved by finite products (see {{mathse:5122756}}).
+- This property is preserved by arbitrary disjoint unions.
diff --git a/properties/P000211.md b/properties/P000211.md
index f358bce0b8..6e7147b500 100644
--- a/properties/P000211.md
+++ b/properties/P000211.md
@@ -37,3 +37,4 @@ This property was introduced by Nyikos in {{zb:0774.54019}}.
 
 - This property is hereditary.
 - This property is preserved by finite products (see {{mathse:5122756}}).
+- This property is preserved by arbitrary disjoint unions.
diff --git a/properties/P000212.md b/properties/P000212.md
index 5a77e7cc44..08c4068626 100644
--- a/properties/P000212.md
+++ b/properties/P000212.md
@@ -37,3 +37,4 @@ are due to Arkhangel'skii ({{zb:0275.54004}}).
 
 - This property is hereditary.
 - This property is preserved by finite products (see {{mathse:5122756}}).
+- This property is preserved by arbitrary disjoint unions.
diff --git a/properties/P000213.md b/properties/P000213.md
index 6bce164dcf..d42d82e5f6 100644
--- a/properties/P000213.md
+++ b/properties/P000213.md
@@ -37,3 +37,4 @@ are due to Arkhangel'skii ({{zb:0275.54004}}).
 
 - This property is hereditary.
 - This property is preserved by finite products (see {{mathse:5122756}}).
+- This property is preserved by arbitrary disjoint unions.
diff --git a/properties/P000214.md b/properties/P000214.md
index 47a4b1e41b..8e56fc5d09 100644
--- a/properties/P000214.md
+++ b/properties/P000214.md
@@ -34,3 +34,4 @@ are due to Arkhangel'skii ({{zb:0275.54004}}).
 #### Meta-properties
 
 - This property is hereditary.
+- This property is preserved by arbitrary disjoint unions.

From 8837c58a47bac00c4e500a32f4b6d2603579e0d6 Mon Sep 17 00:00:00 2001
From: Patrick Rabau <70125716+prabau@users.noreply.github.com>
Date: Sun, 15 Feb 2026 19:38:31 -0500
Subject: [PATCH 4/6] metaprop for Kolmogorov quotient

---
 properties/P000210.md | 1 +
 properties/P000211.md | 1 +
 properties/P000212.md | 1 +
 properties/P000213.md | 1 +
 properties/P000214.md | 1 +
 5 files changed, 5 insertions(+)

diff --git a/properties/P000210.md b/properties/P000210.md
index a371af8414..6326f39e9a 100644
--- a/properties/P000210.md
+++ b/properties/P000210.md
@@ -37,5 +37,6 @@ are due to Arkhangel'skii ({{zb:0275.54004}}).
 #### Meta-properties
 
 - This property is hereditary.
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
 - This property is preserved by finite products (see {{mathse:5122756}}).
 - This property is preserved by arbitrary disjoint unions.
diff --git a/properties/P000211.md b/properties/P000211.md
index 6e7147b500..780e51f547 100644
--- a/properties/P000211.md
+++ b/properties/P000211.md
@@ -36,5 +36,6 @@ This property was introduced by Nyikos in {{zb:0774.54019}}.
 #### Meta-properties
 
 - This property is hereditary.
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
 - This property is preserved by finite products (see {{mathse:5122756}}).
 - This property is preserved by arbitrary disjoint unions.
diff --git a/properties/P000212.md b/properties/P000212.md
index 08c4068626..286ef9318b 100644
--- a/properties/P000212.md
+++ b/properties/P000212.md
@@ -36,5 +36,6 @@ are due to Arkhangel'skii ({{zb:0275.54004}}).
 #### Meta-properties
 
 - This property is hereditary.
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
 - This property is preserved by finite products (see {{mathse:5122756}}).
 - This property is preserved by arbitrary disjoint unions.
diff --git a/properties/P000213.md b/properties/P000213.md
index d42d82e5f6..e0d5ca8210 100644
--- a/properties/P000213.md
+++ b/properties/P000213.md
@@ -36,5 +36,6 @@ are due to Arkhangel'skii ({{zb:0275.54004}}).
 #### Meta-properties
 
 - This property is hereditary.
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
 - This property is preserved by finite products (see {{mathse:5122756}}).
 - This property is preserved by arbitrary disjoint unions.
diff --git a/properties/P000214.md b/properties/P000214.md
index 8e56fc5d09..dd25d3dbda 100644
--- a/properties/P000214.md
+++ b/properties/P000214.md
@@ -34,4 +34,5 @@ are due to Arkhangel'skii ({{zb:0275.54004}}).
 #### Meta-properties
 
 - This property is hereditary.
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
 - This property is preserved by arbitrary disjoint unions.

From 66e2fedbf6cdd4d0b3bd5b7a4a847aefdcbc193c Mon Sep 17 00:00:00 2001
From: Patrick Rabau <70125716+prabau@users.noreply.github.com>
Date: Sun, 15 Feb 2026 23:16:30 -0500
Subject: [PATCH 5/6] refs for Kolmogorov quotient

---
 properties/P000210.md | 4 +++-
 properties/P000211.md | 4 +++-
 properties/P000212.md | 4 +++-
 properties/P000213.md | 4 +++-
 properties/P000214.md | 4 +++-
 5 files changed, 15 insertions(+), 5 deletions(-)

diff --git a/properties/P000210.md b/properties/P000210.md
index 6326f39e9a..0479367145 100644
--- a/properties/P000210.md
+++ b/properties/P000210.md
@@ -12,6 +12,8 @@ refs:
     name: The frequency spectrum of a topological space and the classification of spaces (A. V. Arkhangel'skii)
   - mathse: 5122756
     name: Are the Arkhangel'skii $\alpha_i$ properties preserved by finite products?
+  - mathse: 5123855
+    name: Does the Arkhangel'skii $\alpha_1$ property hold for $X$ if it holds for its Kolmogorov quotient?
 ---
 
 For every $x\in X$ and every sequence of countably infinite sets $S_1,S_2,\dots\subseteq X$ with
@@ -37,6 +39,6 @@ are due to Arkhangel'skii ({{zb:0275.54004}}).
 #### Meta-properties
 
 - This property is hereditary.
-- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does (see {{mathse:5123855}}).
 - This property is preserved by finite products (see {{mathse:5122756}}).
 - This property is preserved by arbitrary disjoint unions.
diff --git a/properties/P000211.md b/properties/P000211.md
index 780e51f547..454ba409f7 100644
--- a/properties/P000211.md
+++ b/properties/P000211.md
@@ -10,6 +10,8 @@ refs:
     name: Subsets of ${}^\omega\omega$ and the Fréchet-Urysohn and $\alpha_i$-properties (P. Nyikos)
   - mathse: 5122756
     name: Are the Arkhangel'skii $\alpha_i$ properties preserved by finite products?
+  - mathse: 5123855
+    name: Does the Arkhangel'skii $\alpha_1$ property hold for $X$ if it holds for its Kolmogorov quotient?
 ---
 
 For every $x\in X$ and every sequence of pairwise disjoint countably infinite sets
@@ -36,6 +38,6 @@ This property was introduced by Nyikos in {{zb:0774.54019}}.
 #### Meta-properties
 
 - This property is hereditary.
-- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does (see {{mathse:5123855}}).
 - This property is preserved by finite products (see {{mathse:5122756}}).
 - This property is preserved by arbitrary disjoint unions.
diff --git a/properties/P000212.md b/properties/P000212.md
index 286ef9318b..bb6888a7a8 100644
--- a/properties/P000212.md
+++ b/properties/P000212.md
@@ -12,6 +12,8 @@ refs:
     name: The frequency spectrum of a topological space and the classification of spaces (A. V. Arkhangel'skii)
   - mathse: 5122756
     name: Are the Arkhangel'skii $\alpha_i$ properties preserved by finite products?
+  - mathse: 5123855
+    name: Does the Arkhangel'skii $\alpha_1$ property hold for $X$ if it holds for its Kolmogorov quotient?
 ---
 
 For every $x\in X$ and every sequence of countably infinite sets $S_1,S_2,\dots\subseteq X$ with
@@ -36,6 +38,6 @@ are due to Arkhangel'skii ({{zb:0275.54004}}).
 #### Meta-properties
 
 - This property is hereditary.
-- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does (see {{mathse:5123855}}).
 - This property is preserved by finite products (see {{mathse:5122756}}).
 - This property is preserved by arbitrary disjoint unions.
diff --git a/properties/P000213.md b/properties/P000213.md
index e0d5ca8210..870038a99e 100644
--- a/properties/P000213.md
+++ b/properties/P000213.md
@@ -12,6 +12,8 @@ refs:
     name: The frequency spectrum of a topological space and the classification of spaces (A. V. Arkhangel'skii)
   - mathse: 5122756
     name: Are the Arkhangel'skii $\alpha_i$ properties preserved by finite products?
+  - mathse: 5123855
+    name: Does the Arkhangel'skii $\alpha_1$ property hold for $X$ if it holds for its Kolmogorov quotient?
 ---
 
 For every $x\in X$ and every sequence of countably infinite sets $S_1,S_2,\dots\subseteq X$ with
@@ -36,6 +38,6 @@ are due to Arkhangel'skii ({{zb:0275.54004}}).
 #### Meta-properties
 
 - This property is hereditary.
-- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does (see {{mathse:5123855}}).
 - This property is preserved by finite products (see {{mathse:5122756}}).
 - This property is preserved by arbitrary disjoint unions.
diff --git a/properties/P000214.md b/properties/P000214.md
index dd25d3dbda..44ffd0b3fd 100644
--- a/properties/P000214.md
+++ b/properties/P000214.md
@@ -10,6 +10,8 @@ refs:
     name: Subsets of ${}^\omega\omega$ and the Fréchet-Urysohn and $\alpha_i$-properties (P. Nyikos)
   - zb: "0275.54004"
     name: The frequency spectrum of a topological space and the classification of spaces (A. V. Arkhangel'skii)
+  - mathse: 5123855
+    name: Does the Arkhangel'skii $\alpha_1$ property hold for $X$ if it holds for its Kolmogorov quotient?
 ---
 
 For every $x\in X$ and every sequence of countably infinite sets $S_1,S_2,\dots\subseteq X$ with
@@ -34,5 +36,5 @@ are due to Arkhangel'skii ({{zb:0275.54004}}).
 #### Meta-properties
 
 - This property is hereditary.
-- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does.
+- $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does (see {{mathse:5123855}}).
 - This property is preserved by arbitrary disjoint unions.

From c85a91a07444d75cbd7b0d7f217a21189b36dcde Mon Sep 17 00:00:00 2001
From: Patrick Rabau <70125716+prabau@users.noreply.github.com>
Date: Wed, 4 Mar 2026 02:39:42 -0500
Subject: [PATCH 6/6] countable products; and S181 trait

---
 properties/P000210.md                | 6 +++---
 properties/P000211.md                | 6 +++---
 properties/P000212.md                | 6 +++---
 properties/P000213.md                | 6 +++---
 spaces/S000181/properties/P000210.md | 8 ++++++++
 5 files changed, 20 insertions(+), 12 deletions(-)
 create mode 100644 spaces/S000181/properties/P000210.md

diff --git a/properties/P000210.md b/properties/P000210.md
index 0479367145..ef0ab1984b 100644
--- a/properties/P000210.md
+++ b/properties/P000210.md
@@ -10,8 +10,8 @@ refs:
     name: Subsets of ${}^\omega\omega$ and the Fréchet-Urysohn and $\alpha_i$-properties (P. Nyikos)
   - zb: "0275.54004"
     name: The frequency spectrum of a topological space and the classification of spaces (A. V. Arkhangel'skii)
-  - mathse: 5122756
-    name: Are the Arkhangel'skii $\alpha_i$ properties preserved by finite products?
+  - mathse: 5127008
+    name: Answer to "Are the Arkhangel'skii $\alpha_i$ properties preserved by finite products?"
   - mathse: 5123855
     name: Does the Arkhangel'skii $\alpha_1$ property hold for $X$ if it holds for its Kolmogorov quotient?
 ---
@@ -40,5 +40,5 @@ are due to Arkhangel'skii ({{zb:0275.54004}}).
 
 - This property is hereditary.
 - $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does (see {{mathse:5123855}}).
-- This property is preserved by finite products (see {{mathse:5122756}}).
+- This property is preserved by countable products (see {{mathse:5127008}}).
 - This property is preserved by arbitrary disjoint unions.
diff --git a/properties/P000211.md b/properties/P000211.md
index 454ba409f7..26f257abc3 100644
--- a/properties/P000211.md
+++ b/properties/P000211.md
@@ -8,8 +8,8 @@ refs:
     name: Arhangel'skii sheaf amalgamations in topological groups (Tsaban & Zdomskyy)
   - zb: "0774.54019"
     name: Subsets of ${}^\omega\omega$ and the Fréchet-Urysohn and $\alpha_i$-properties (P. Nyikos)
-  - mathse: 5122756
-    name: Are the Arkhangel'skii $\alpha_i$ properties preserved by finite products?
+  - mathse: 5127008
+    name: Answer to "Are the Arkhangel'skii $\alpha_i$ properties preserved by finite products?"
   - mathse: 5123855
     name: Does the Arkhangel'skii $\alpha_1$ property hold for $X$ if it holds for its Kolmogorov quotient?
 ---
@@ -39,5 +39,5 @@ This property was introduced by Nyikos in {{zb:0774.54019}}.
 
 - This property is hereditary.
 - $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does (see {{mathse:5123855}}).
-- This property is preserved by finite products (see {{mathse:5122756}}).
+- This property is preserved by countable products (see {{mathse:5127008}}).
 - This property is preserved by arbitrary disjoint unions.
diff --git a/properties/P000212.md b/properties/P000212.md
index bb6888a7a8..01dbae50b7 100644
--- a/properties/P000212.md
+++ b/properties/P000212.md
@@ -10,8 +10,8 @@ refs:
     name: Subsets of ${}^\omega\omega$ and the Fréchet-Urysohn and $\alpha_i$-properties (P. Nyikos)
   - zb: "0275.54004"
     name: The frequency spectrum of a topological space and the classification of spaces (A. V. Arkhangel'skii)
-  - mathse: 5122756
-    name: Are the Arkhangel'skii $\alpha_i$ properties preserved by finite products?
+  - mathse: 5127008
+    name: Answer to "Are the Arkhangel'skii $\alpha_i$ properties preserved by finite products?"
   - mathse: 5123855
     name: Does the Arkhangel'skii $\alpha_1$ property hold for $X$ if it holds for its Kolmogorov quotient?
 ---
@@ -39,5 +39,5 @@ are due to Arkhangel'skii ({{zb:0275.54004}}).
 
 - This property is hereditary.
 - $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does (see {{mathse:5123855}}).
-- This property is preserved by finite products (see {{mathse:5122756}}).
+- This property is preserved by countable products (see {{mathse:5127008}}).
 - This property is preserved by arbitrary disjoint unions.
diff --git a/properties/P000213.md b/properties/P000213.md
index 870038a99e..4d5b72ac66 100644
--- a/properties/P000213.md
+++ b/properties/P000213.md
@@ -10,8 +10,8 @@ refs:
     name: Subsets of ${}^\omega\omega$ and the Fréchet-Urysohn and $\alpha_i$-properties (P. Nyikos)
   - zb: "0275.54004"
     name: The frequency spectrum of a topological space and the classification of spaces (A. V. Arkhangel'skii)
-  - mathse: 5122756
-    name: Are the Arkhangel'skii $\alpha_i$ properties preserved by finite products?
+  - mathse: 5127008
+    name: Answer to "Are the Arkhangel'skii $\alpha_i$ properties preserved by finite products?"
   - mathse: 5123855
     name: Does the Arkhangel'skii $\alpha_1$ property hold for $X$ if it holds for its Kolmogorov quotient?
 ---
@@ -39,5 +39,5 @@ are due to Arkhangel'skii ({{zb:0275.54004}}).
 
 - This property is hereditary.
 - $X$ satisfies this property iff its Kolmogorov quotient $\text{Kol}(X)$ does (see {{mathse:5123855}}).
-- This property is preserved by finite products (see {{mathse:5122756}}).
+- This property is preserved by countable products (see {{mathse:5127008}}).
 - This property is preserved by arbitrary disjoint unions.
diff --git a/spaces/S000181/properties/P000210.md b/spaces/S000181/properties/P000210.md
new file mode 100644
index 0000000000..a40f9cd4dd
--- /dev/null
+++ b/spaces/S000181/properties/P000210.md
@@ -0,0 +1,8 @@
+---
+space: S000181
+property: P000210
+value: true
+---
+
+$X$ is a subspace of a countable product of {P210} spaces,
+since {S36|P210}.