From 166666521de791f22b2ae56ffb6cb2bd1cb97a11 Mon Sep 17 00:00:00 2001
From: yhx-12243 <yhx12243@gmail.com>
Date: Tue, 20 Jan 2026 14:46:07 +0800
Subject: [PATCH 01/15] =?UTF-8?q?Has=20closed=20discrete=20subset=20of=20s?=
 =?UTF-8?q?ize=20=F0=9D=94=A0=20(part=204)?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 properties/P000227.md                |  1 +
 spaces/S000077/properties/P000227.md |  7 +++++++
 spaces/S000081/properties/P000198.md |  8 ++++++++
 spaces/S000101/properties/P000227.md | 12 ++++++++++++
 spaces/S000107/properties/P000227.md |  7 +++++++
 spaces/S000136/properties/P000227.md |  7 +++++++
 spaces/S000137/properties/P000227.md |  7 +++++++
 spaces/S001103/properties/P000227.md |  7 +++++++
 8 files changed, 56 insertions(+)
 create mode 100644 spaces/S000077/properties/P000227.md
 create mode 100644 spaces/S000081/properties/P000198.md
 create mode 100644 spaces/S000101/properties/P000227.md
 create mode 100644 spaces/S000107/properties/P000227.md
 create mode 100644 spaces/S000136/properties/P000227.md
 create mode 100644 spaces/S000137/properties/P000227.md
 create mode 100644 spaces/S001103/properties/P000227.md

diff --git a/properties/P000227.md b/properties/P000227.md
index 418aff8dc..af105e728 100644
--- a/properties/P000227.md
+++ b/properties/P000227.md
@@ -22,3 +22,4 @@ Compare with these properties, where $D$ denotes a discrete closed set in $X$:
 #### Meta-properties
 
 - This property is preserved in any finer topology.
+- If a closed subspace of $X$ satisfies this property, so does $X$.
diff --git a/spaces/S000077/properties/P000227.md b/spaces/S000077/properties/P000227.md
new file mode 100644
index 000000000..064535fd2
--- /dev/null
+++ b/spaces/S000077/properties/P000227.md
@@ -0,0 +1,7 @@
+---
+space: S000077
+property: P000227
+value: true
+---
+
+$X$ contains a closed subspace {S63}, and {S63|P227}.
diff --git a/spaces/S000081/properties/P000198.md b/spaces/S000081/properties/P000198.md
new file mode 100644
index 000000000..cb5e6ff81
--- /dev/null
+++ b/spaces/S000081/properties/P000198.md
@@ -0,0 +1,8 @@
+---
+space: S000081
+property: P000198
+value: true
+---
+
+$X$ is the union of $[0, \omega_1] \times [-1, 0)$, $[0, \omega_1] \times (0, 1]$, $[0, \omega_1) \times \{0\}$ and $\left\{ \left< \omega_1, 0 \right> \right\}$.
+All of them are {P198}.
diff --git a/spaces/S000101/properties/P000227.md b/spaces/S000101/properties/P000227.md
new file mode 100644
index 000000000..afb35a02c
--- /dev/null
+++ b/spaces/S000101/properties/P000227.md
@@ -0,0 +1,12 @@
+---
+space: S000101
+property: P000227
+value: true
+refs:
+  - zb: "0684.54001"
+    name: General Topology (Engelking, 1989)
+---
+
+See Exercise 3.1.H(a) in {{zb:0684.54001}}.
+
+See also https://dantopology.wordpress.com/2014/03/08/looking-for-a-closed-and-discrete-subspace-of-a-product-space/.
diff --git a/spaces/S000107/properties/P000227.md b/spaces/S000107/properties/P000227.md
new file mode 100644
index 000000000..118a3aa91
--- /dev/null
+++ b/spaces/S000107/properties/P000227.md
@@ -0,0 +1,7 @@
+---
+space: S000107
+property: P000227
+value: true
+---
+
+$\left\{ 0, 1 \right\}^\omega$ is a closed and discrete subset of cardinality $\mathfrak c$.
diff --git a/spaces/S000136/properties/P000227.md b/spaces/S000136/properties/P000227.md
new file mode 100644
index 000000000..76fceda93
--- /dev/null
+++ b/spaces/S000136/properties/P000227.md
@@ -0,0 +1,7 @@
+---
+space: S000136
+property: P000227
+value: true
+---
+
+$M$ is a closed and discrete subset of cardinality $\mathfrak c$.
diff --git a/spaces/S000137/properties/P000227.md b/spaces/S000137/properties/P000227.md
new file mode 100644
index 000000000..e60dd2992
--- /dev/null
+++ b/spaces/S000137/properties/P000227.md
@@ -0,0 +1,7 @@
+---
+space: S000137
+property: P000227
+value: true
+---
+
+$M$ is a closed and discrete subset of cardinality $\mathfrak c$.
diff --git a/spaces/S001103/properties/P000227.md b/spaces/S001103/properties/P000227.md
new file mode 100644
index 000000000..3a2d36cd5
--- /dev/null
+++ b/spaces/S001103/properties/P000227.md
@@ -0,0 +1,7 @@
+---
+space: S001103
+property: P000227
+value: true
+---
+
+$X$ contains a closed subspace {S101}, and {S101|P227}.

From 6d94c56c7de115ecfb1c0b29e8f48f7cc653d8cc Mon Sep 17 00:00:00 2001
From: yhx-12243 <yhx12243@gmail.com>
Date: Tue, 20 Jan 2026 15:20:30 +0800
Subject: [PATCH 02/15] fix link in S101

---
 spaces/S000101/properties/P000227.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/spaces/S000101/properties/P000227.md b/spaces/S000101/properties/P000227.md
index afb35a02c..f968f8b8e 100644
--- a/spaces/S000101/properties/P000227.md
+++ b/spaces/S000101/properties/P000227.md
@@ -9,4 +9,4 @@ refs:
 
 See Exercise 3.1.H(a) in {{zb:0684.54001}}.
 
-See also https://dantopology.wordpress.com/2014/03/08/looking-for-a-closed-and-discrete-subspace-of-a-product-space/.
+See also <https://dantopology.wordpress.com/2014/03/08/looking-for-a-closed-and-discrete-subspace-of-a-product-space/>.

From ec9d6538e253167efb6c6c9c63fe2abc476bcdd6 Mon Sep 17 00:00:00 2001
From: yhx-12243 <yhx12243@gmail.com>
Date: Thu, 22 Jan 2026 10:28:20 +0800
Subject: [PATCH 03/15] Apply suggestions from code review

Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com>
---
 spaces/S000077/properties/P000227.md | 2 +-
 spaces/S001103/properties/P000227.md | 2 +-
 2 files changed, 2 insertions(+), 2 deletions(-)

diff --git a/spaces/S000077/properties/P000227.md b/spaces/S000077/properties/P000227.md
index 064535fd2..56bc780a4 100644
--- a/spaces/S000077/properties/P000227.md
+++ b/spaces/S000077/properties/P000227.md
@@ -4,4 +4,4 @@ property: P000227
 value: true
 ---
 
-$X$ contains a closed subspace {S63}, and {S63|P227}.
+$X$ contains {S63} as a closed subspace, and {S63|P227}.
diff --git a/spaces/S001103/properties/P000227.md b/spaces/S001103/properties/P000227.md
index 3a2d36cd5..030747c33 100644
--- a/spaces/S001103/properties/P000227.md
+++ b/spaces/S001103/properties/P000227.md
@@ -4,4 +4,4 @@ property: P000227
 value: true
 ---
 
-$X$ contains a closed subspace {S101}, and {S101|P227}.
+$X$ contains a closed subspace homeomorphic to {S101}, and {S101|P227}.

From 4943288d203a73e318412a177c602f7ff37a7dc3 Mon Sep 17 00:00:00 2001
From: yhx-12243 <yhx12243@gmail.com>
Date: Thu, 22 Jan 2026 10:30:51 +0800
Subject: [PATCH 04/15] P198 meta-properties

---
 properties/P000198.md                | 5 +++++
 spaces/S000079/properties/P000198.md | 6 +-----
 spaces/S000153/properties/P000198.md | 9 +++++++++
 3 files changed, 15 insertions(+), 5 deletions(-)
 create mode 100644 spaces/S000153/properties/P000198.md

diff --git a/properties/P000198.md b/properties/P000198.md
index 081c8d723..51cb27f22 100644
--- a/properties/P000198.md
+++ b/properties/P000198.md
@@ -16,3 +16,8 @@ That is, every closed discrete subspace of $X$ is countable.
 Equivalently, every uncountable set $S\subseteq X$ has a limit point in $X$.
 
 See {{mr:0776620}} or section a-3 of {{doi:10.1016/B978-0-444-50355-8.X5000-4}}.
+
+----
+#### Meta-properties
+
+- This property is preserved by countable (not necessarily disjoint) unions.
diff --git a/spaces/S000079/properties/P000198.md b/spaces/S000079/properties/P000198.md
index dde83cd59..f66a60bb7 100644
--- a/spaces/S000079/properties/P000198.md
+++ b/spaces/S000079/properties/P000198.md
@@ -7,8 +7,4 @@ value: true
 $X$ is a countable union of subspaces
 $[0,\omega_1]\times\{n\}$ for $n<\omega$ and $[0,\omega_1)\times\{\omega\}$ which are homeomorphic to {S36}
 and {S35}, respectively.
-Every closed and discrete subset of $X$ has
-countable intersection with each of them
-({S36|P198}
- and {S35|P198})
-and therefore is countable itself.
+Both of them are {P198}.
diff --git a/spaces/S000153/properties/P000198.md b/spaces/S000153/properties/P000198.md
new file mode 100644
index 000000000..9630355d3
--- /dev/null
+++ b/spaces/S000153/properties/P000198.md
@@ -0,0 +1,9 @@
+---
+space: S000153
+property: P000198
+value: true
+---
+
+$X$ is the union of $\{0\} \times (0, 1)$ and $\left[ 1, \omega_1 \right) \times [0, 1)$,
+which are homeomorphic to {S25} and {S39}, respectively.
+Both of them are {P198}.

From 293d34ff2410d2a1e173377770d7877e0fe29eda Mon Sep 17 00:00:00 2001
From: yhx-12243 <yhx12243@gmail.com>
Date: Fri, 23 Jan 2026 16:56:51 +0800
Subject: [PATCH 05/15] remove redundant

---
 spaces/S000107/properties/P000017.md | 10 ----------
 spaces/S000107/properties/P000018.md | 11 -----------
 2 files changed, 21 deletions(-)
 delete mode 100644 spaces/S000107/properties/P000017.md
 delete mode 100644 spaces/S000107/properties/P000018.md

diff --git a/spaces/S000107/properties/P000017.md b/spaces/S000107/properties/P000017.md
deleted file mode 100644
index 84d2fc1d5..000000000
--- a/spaces/S000107/properties/P000017.md
+++ /dev/null
@@ -1,10 +0,0 @@
----
-space: S000107
-property: P000017
-value: false
-refs:
-- zb: "0386.54001"
-  name: Counterexamples in Topology
----
-
-See item #4 for space #109 in {{zb:0386.54001}}.
diff --git a/spaces/S000107/properties/P000018.md b/spaces/S000107/properties/P000018.md
deleted file mode 100644
index c6a1b614b..000000000
--- a/spaces/S000107/properties/P000018.md
+++ /dev/null
@@ -1,11 +0,0 @@
----
-space: S000107
-property: P000018
-value: false
-refs:
-- zb: "0386.54001"
-  name: Counterexamples in Topology
----
-
-Asserted in the General Reference Chart for space #109 in
-{{zb:0386.54001}}.

From 9c4505fcf8de73308ba518a2773bacd100a1083a Mon Sep 17 00:00:00 2001
From: yhx-12243 <yhx12243@gmail.com>
Date: Fri, 23 Jan 2026 16:57:41 +0800
Subject: [PATCH 06/15] remove redundant S136

---
 spaces/S000136/properties/P000018.md | 11 -----------
 1 file changed, 11 deletions(-)
 delete mode 100644 spaces/S000136/properties/P000018.md

diff --git a/spaces/S000136/properties/P000018.md b/spaces/S000136/properties/P000018.md
deleted file mode 100644
index ad5f45721..000000000
--- a/spaces/S000136/properties/P000018.md
+++ /dev/null
@@ -1,11 +0,0 @@
----
-space: S000136
-property: P000018
-value: false
-refs:
-- zb: "0386.54001"
-  name: Counterexamples in Topology
----
-
-Asserted in the General Reference Chart for space #142 in
-{{zb:0386.54001}}.

From a1c9aff70ebfe1387c9f0bf4106b774a58029a02 Mon Sep 17 00:00:00 2001
From: yhx-12243 <yhx12243@gmail.com>
Date: Fri, 23 Jan 2026 16:59:34 +0800
Subject: [PATCH 07/15] remove redundant S101

---
 spaces/S000101/properties/P000013.md | 11 -----------
 spaces/S000101/properties/P000017.md | 11 -----------
 2 files changed, 22 deletions(-)
 delete mode 100644 spaces/S000101/properties/P000013.md
 delete mode 100644 spaces/S000101/properties/P000017.md

diff --git a/spaces/S000101/properties/P000013.md b/spaces/S000101/properties/P000013.md
deleted file mode 100644
index ed88b1096..000000000
--- a/spaces/S000101/properties/P000013.md
+++ /dev/null
@@ -1,11 +0,0 @@
----
-space: S000101
-property: P000013
-value: false
-refs:
-- zb: "0386.54001"
-  name: Counterexamples in Topology
----
-
-Asserted in the General Reference Chart for space #103
-in {{doi:10.1007\/978-1-4612-6290-9_6}}.
diff --git a/spaces/S000101/properties/P000017.md b/spaces/S000101/properties/P000017.md
deleted file mode 100644
index 7b4ba081a..000000000
--- a/spaces/S000101/properties/P000017.md
+++ /dev/null
@@ -1,11 +0,0 @@
----
-space: S000101
-property: P000017
-value: false
-refs:
-- zb: "0386.54001"
-  name: Counterexamples in Topology
----
-
-Asserted in the General Reference Chart for space #103
-in {{doi:10.1007\/978-1-4612-6290-9_6}}.

From aaa3fb03452261df60c640b82f11a0a3c7623f6d Mon Sep 17 00:00:00 2001
From: Patrick Rabau <70125716+prabau@users.noreply.github.com>
Date: Sat, 24 Jan 2026 18:16:31 -0500
Subject: [PATCH 08/15] S79-P198 rephrase

---
 spaces/S000079/properties/P000198.md | 9 +++++----
 1 file changed, 5 insertions(+), 4 deletions(-)

diff --git a/spaces/S000079/properties/P000198.md b/spaces/S000079/properties/P000198.md
index f66a60bb7..c895dd474 100644
--- a/spaces/S000079/properties/P000198.md
+++ b/spaces/S000079/properties/P000198.md
@@ -4,7 +4,8 @@ property: P000198
 value: true
 ---
 
-$X$ is a countable union of subspaces
-$[0,\omega_1]\times\{n\}$ for $n<\omega$ and $[0,\omega_1)\times\{\omega\}$ which are homeomorphic to {S36}
-and {S35}, respectively.
-Both of them are {P198}.
+$X$ is covered by the countably many subspaces
+$[0,\omega_1]\times\{n\}$ for $n<\omega$ and $[0,\omega_1)\times\{\omega\}$,
+with each of them {P198}
+since {S36|P198}
+and {S35|P198}.

From 68ee77cb27888db291697f76923f2e0c1bb334e6 Mon Sep 17 00:00:00 2001
From: yhx-12243 <yhx12243@gmail.com>
Date: Sun, 25 Jan 2026 08:32:30 +0800
Subject: [PATCH 09/15] Apply suggestions from code review

Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com>
---
 properties/P000198.md                | 2 +-
 spaces/S000153/properties/P000198.md | 6 +++---
 2 files changed, 4 insertions(+), 4 deletions(-)

diff --git a/properties/P000198.md b/properties/P000198.md
index 51cb27f22..c52212bb3 100644
--- a/properties/P000198.md
+++ b/properties/P000198.md
@@ -20,4 +20,4 @@ See {{mr:0776620}} or section a-3 of {{doi:10.1016/B978-0-444-50355-8.X5000-4}}.
 ----
 #### Meta-properties
 
-- This property is preserved by countable (not necessarily disjoint) unions.
+- If $X$ is covered by countably many subspaces, each having the property, then so does $X$.
diff --git a/spaces/S000153/properties/P000198.md b/spaces/S000153/properties/P000198.md
index 9630355d3..655df18d7 100644
--- a/spaces/S000153/properties/P000198.md
+++ b/spaces/S000153/properties/P000198.md
@@ -4,6 +4,6 @@ property: P000198
 value: true
 ---
 
-$X$ is the union of $\{0\} \times (0, 1)$ and $\left[ 1, \omega_1 \right) \times [0, 1)$,
-which are homeomorphic to {S25} and {S39}, respectively.
-Both of them are {P198}.
+$X$ is the union of the subspaces $\{0\} \times (0, 1)$ and $\left[ 1, \omega_1 \right) \times [0, 1)$,
+which are homeomorphic to {S25} and {S38}, respectively.
+Each has countable extent, as {S25|P198} and {S38|P198}.

From 0c0b1b01a1c854787411813a52f83e8b419154ef Mon Sep 17 00:00:00 2001
From: yhx-12243 <yhx12243@gmail.com>
Date: Sun, 25 Jan 2026 08:37:08 +0800
Subject: [PATCH 10/15] Add reference of S&S

---
 spaces/S000136/properties/P000227.md | 5 +++++
 spaces/S000137/properties/P000227.md | 5 +++++
 2 files changed, 10 insertions(+)

diff --git a/spaces/S000136/properties/P000227.md b/spaces/S000136/properties/P000227.md
index 76fceda93..fc93cbdd8 100644
--- a/spaces/S000136/properties/P000227.md
+++ b/spaces/S000136/properties/P000227.md
@@ -2,6 +2,11 @@
 space: S000136
 property: P000227
 value: true
+refs:
+- zb: "0386.54001"
+  name: Counterexamples in Topology
 ---
 
 $M$ is a closed and discrete subset of cardinality $\mathfrak c$.
+
+See item #1 for space #142 in {{zb:0386.54001}}.
diff --git a/spaces/S000137/properties/P000227.md b/spaces/S000137/properties/P000227.md
index e60dd2992..4e7cc4197 100644
--- a/spaces/S000137/properties/P000227.md
+++ b/spaces/S000137/properties/P000227.md
@@ -2,6 +2,11 @@
 space: S000137
 property: P000227
 value: true
+refs:
+- zb: "0386.54001"
+  name: Counterexamples in Topology
 ---
 
 $M$ is a closed and discrete subset of cardinality $\mathfrak c$.
+
+See item #1 for space #143 in {{zb:0386.54001}}.

From 1fb0704add3af1bcc5f9e67042cb59764447b194 Mon Sep 17 00:00:00 2001
From: yhx-12243 <yhx12243@gmail.com>
Date: Sun, 25 Jan 2026 08:49:15 +0800
Subject: [PATCH 11/15] Explain S81

---
 spaces/S000081/properties/P000198.md | 5 ++++-
 1 file changed, 4 insertions(+), 1 deletion(-)

diff --git a/spaces/S000081/properties/P000198.md b/spaces/S000081/properties/P000198.md
index cb5e6ff81..3aba5aad7 100644
--- a/spaces/S000081/properties/P000198.md
+++ b/spaces/S000081/properties/P000198.md
@@ -5,4 +5,7 @@ value: true
 ---
 
 $X$ is the union of $[0, \omega_1] \times [-1, 0)$, $[0, \omega_1] \times (0, 1]$, $[0, \omega_1) \times \{0\}$ and $\left\{ \left< \omega_1, 0 \right> \right\}$.
-All of them are {P198}.
+All of them are {P198}:
+* The first two space is a product of a {P16} space and {P18} space. So it is {P18} and {T562}.
+* $[0, \omega_1) \times \{0\}$ is homeomorphic to {S35} and {S35|P198}.
+* The last space is {S162} and {S162|P198}.

From aef6e450288942caacec6fa70eb9a0ac994f62ed Mon Sep 17 00:00:00 2001
From: yhx-12243 <yhx12243@gmail.com>
Date: Sun, 25 Jan 2026 11:03:36 +0800
Subject: [PATCH 12/15] Update spaces/S000081/properties/P000198.md

Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com>
---
 spaces/S000081/properties/P000198.md | 10 +++++-----
 1 file changed, 5 insertions(+), 5 deletions(-)

diff --git a/spaces/S000081/properties/P000198.md b/spaces/S000081/properties/P000198.md
index 3aba5aad7..6d5b80616 100644
--- a/spaces/S000081/properties/P000198.md
+++ b/spaces/S000081/properties/P000198.md
@@ -4,8 +4,8 @@ property: P000198
 value: true
 ---
 
-$X$ is the union of $[0, \omega_1] \times [-1, 0)$, $[0, \omega_1] \times (0, 1]$, $[0, \omega_1) \times \{0\}$ and $\left\{ \left< \omega_1, 0 \right> \right\}$.
-All of them are {P198}:
-* The first two space is a product of a {P16} space and {P18} space. So it is {P18} and {T562}.
-* $[0, \omega_1) \times \{0\}$ is homeomorphic to {S35} and {S35|P198}.
-* The last space is {S162} and {S162|P198}.
+$X$ is the union of finitely many subspaces, each with {P198}:
+* $[0, \omega_1] \times [-1, 0)$, because it is {P18} as the product of a {P16} space and a {P18} space, and {T562}.
+* $[0, \omega_1] \times (0, 1]$ for the same reason.
+* $[0, \omega_1) \times \{0\}$, because it is homeomorphic to {S35} and {S35|P198}.
+* $\{\left< \omega_1, 0 \right>\}$, because {S162|P198}.

From ef1d2bed9d5e9b73715498d02f5fc69dc728b7bc Mon Sep 17 00:00:00 2001
From: yhx-12243 <yhx12243@gmail.com>
Date: Mon, 26 Jan 2026 07:31:08 +0800
Subject: [PATCH 13/15] update S81, S136

---
 spaces/S000081/properties/P000198.md | 5 ++++-
 spaces/S000136/properties/P000227.md | 2 ++
 2 files changed, 6 insertions(+), 1 deletion(-)

diff --git a/spaces/S000081/properties/P000198.md b/spaces/S000081/properties/P000198.md
index 6d5b80616..6ad9504f9 100644
--- a/spaces/S000081/properties/P000198.md
+++ b/spaces/S000081/properties/P000198.md
@@ -2,10 +2,13 @@
 space: S000081
 property: P000198
 value: true
+refs:
+- mathse: 2157230
+  name: (Proof) Product of Lindelöf and compact set is Lindelöf
 ---
 
 $X$ is the union of finitely many subspaces, each with {P198}:
-* $[0, \omega_1] \times [-1, 0)$, because it is {P18} as the product of a {P16} space and a {P18} space, and {T562}.
+* $[0, \omega_1] \times [-1, 0)$, because it is {P18} as the product of a {P16} space and a {P18} space (see for example {{mathse:2157230}}), and {T562}.
 * $[0, \omega_1] \times (0, 1]$ for the same reason.
 * $[0, \omega_1) \times \{0\}$, because it is homeomorphic to {S35} and {S35|P198}.
 * $\{\left< \omega_1, 0 \right>\}$, because {S162|P198}.
diff --git a/spaces/S000136/properties/P000227.md b/spaces/S000136/properties/P000227.md
index fc93cbdd8..169db90e2 100644
--- a/spaces/S000136/properties/P000227.md
+++ b/spaces/S000136/properties/P000227.md
@@ -9,4 +9,6 @@ refs:
 
 $M$ is a closed and discrete subset of cardinality $\mathfrak c$.
 
+Also, $X$ contains {S137} as a closed subspace, and {S137|P227}.
+
 See item #1 for space #142 in {{zb:0386.54001}}.

From bf305a0fe78d9206307851bc21b4024768971653 Mon Sep 17 00:00:00 2001
From: yhx-12243 <yhx12243@gmail.com>
Date: Mon, 26 Jan 2026 09:54:05 +0800
Subject: [PATCH 14/15] Change mathse ref

---
 spaces/S000081/properties/P000198.md | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/spaces/S000081/properties/P000198.md b/spaces/S000081/properties/P000198.md
index 6ad9504f9..0cfc0d3e9 100644
--- a/spaces/S000081/properties/P000198.md
+++ b/spaces/S000081/properties/P000198.md
@@ -3,12 +3,12 @@ space: S000081
 property: P000198
 value: true
 refs:
-- mathse: 2157230
-  name: (Proof) Product of Lindelöf and compact set is Lindelöf
+- mathse: 3652735
+  name: If $X$ is Lindelof and $Y$ is compact, then $X \times Y$ is Lindelof
 ---
 
 $X$ is the union of finitely many subspaces, each with {P198}:
-* $[0, \omega_1] \times [-1, 0)$, because it is {P18} as the product of a {P16} space and a {P18} space (see for example {{mathse:2157230}}), and {T562}.
+* $[0, \omega_1] \times [-1, 0)$, because it is {P18} as the product of a {P16} space and a {P18} space (see for example {{mathse:3652735}}), and {T562}.
 * $[0, \omega_1] \times (0, 1]$ for the same reason.
 * $[0, \omega_1) \times \{0\}$, because it is homeomorphic to {S35} and {S35|P198}.
 * $\{\left< \omega_1, 0 \right>\}$, because {S162|P198}.

From 99c825f3008a31dca37b17118b94612d6b2f2db4 Mon Sep 17 00:00:00 2001
From: yhx-12243 <yhx12243@gmail.com>
Date: Thu, 29 Jan 2026 08:56:41 +0800
Subject: [PATCH 15/15] Rephrase S136|P227

---
 spaces/S000136/properties/P000227.md | 5 +++--
 1 file changed, 3 insertions(+), 2 deletions(-)

diff --git a/spaces/S000136/properties/P000227.md b/spaces/S000136/properties/P000227.md
index b4c9a68b2..e9f618c6d 100644
--- a/spaces/S000136/properties/P000227.md
+++ b/spaces/S000136/properties/P000227.md
@@ -7,8 +7,9 @@ refs:
   name: Counterexamples in Topology
 ---
 
-$M$ is a closed and discrete subset of cardinality $\mathfrak c$.
+$X$ contains {S137} as a closed subspace, and {S137|P227}.
+
+Explicitly, $M$ is a closed and discrete subset of cardinality $\mathfrak c$.
 
 See item #1 for space #142 in {{zb:0386.54001}}.
 
-Also, $X$ contains {S137} as a closed subspace, and {S137|P227}.