More aleph_0 spaces

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 $X.$

spaces/S000139/properties/P000183.md

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+---
+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 $X.$

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)

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}}.