From 4c99660d40d66d6f48b5b0796b75a20c3e2d51ae Mon Sep 17 00:00:00 2001
From: Patrick Rabau <prabau@sbcglobal.net>
Date: Sat, 28 Sep 2024 12:37:03 -0400
Subject: [PATCH] More aleph_0 spaces

---
 spaces/S000131/properties/P000183.md | 14 ++++++++++++++
 spaces/S000139/properties/P000183.md | 10 ++++++++++
 spaces/S000156/README.md             |  2 +-
 spaces/S000156/properties/P000183.md | 10 ++++++++++
 4 files changed, 35 insertions(+), 1 deletion(-)
 create mode 100644 spaces/S000131/properties/P000183.md
 create mode 100644 spaces/S000139/properties/P000183.md
 create mode 100644 spaces/S000156/properties/P000183.md

diff --git a/spaces/S000131/properties/P000183.md b/spaces/S000131/properties/P000183.md
new file mode 100644
index 000000000..7e59be3ad
--- /dev/null
+++ b/spaces/S000131/properties/P000183.md
@@ -0,0 +1,14 @@
+---
+space: S000131
+property: P000183
+value: true
+refs:
+  - mathse: 4844426
+    name: Answer to "Can a Fréchet-Urysohn hemicompact Hausdorff space fail to be locally compact?"
+---
+
+Each spine $C_m=(\{m\}\times\omega)\cup\{\infty\}$ is a closed subspace of $X$ homeomorphic to a convergent sequence ({S20});
+and {S20|P183}.
+And every compact subset of $X$ is contained in the union of a finite number of these spines.
+
+Therefore, taking a countable $k$-network from each of the (countably many) spines and forming their union gives a countable $k$-network for $X.$
diff --git a/spaces/S000139/properties/P000183.md b/spaces/S000139/properties/P000183.md
new file mode 100644
index 000000000..b97344c4a
--- /dev/null
+++ b/spaces/S000139/properties/P000183.md
@@ -0,0 +1,10 @@
+---
+space: S000139
+property: P000183
+value: true
+---
+
+Each of the circles (corresponding to an interval $[n,n+1]$, $n\in\mathbb Z$, with the endpoints identified) is a closed subspace of $X$ and {S170|P183}.
+And every compact subset of $X$ is contained in the union of a finite number of these circles.
+
+Therefore, taking a countable $k$-network from each of the (countably many) circles and forming their union gives a countable $k$-network for $X.$
diff --git a/spaces/S000156/README.md b/spaces/S000156/README.md
index 82e41b3ce..92d1f20fd 100644
--- a/spaces/S000156/README.md
+++ b/spaces/S000156/README.md
@@ -2,7 +2,7 @@
 uid: S000156
 name: Arens space
 aliases:
-- S_2
+- $S_2$
 refs:
   - mr: 37500
     name: Note on convergence in topology (Arens)
diff --git a/spaces/S000156/properties/P000183.md b/spaces/S000156/properties/P000183.md
new file mode 100644
index 000000000..cc8826c16
--- /dev/null
+++ b/spaces/S000156/properties/P000183.md
@@ -0,0 +1,10 @@
+---
+space: S000156
+property: P000183
+value: true
+refs:
+  - mathse: 4673444
+    name: Answer to "Is the Arens space hemicompact but not locally compact?"
+---
+
+See the last part of {{mathse:4673444}}.