Some traits for S174

spaces/S000174/properties/P000023.md

@@ -0,0 +1,11 @@
+---
+space: S000174
+property: P000023
+value: false
+---
+
+At the point $p=(\omega, \omega, \dots)$, the clopen sets
+$\{\omega\}^n \times (\omega+1)^\omega$ form a neighborhood base. Each of
+these sets has a topology finer than the product topology on
+$\{\omega\}^n \times (\omega+1)^\omega$ when $\omega+1$ is discrete. The
+latter space is not compact, so none of these neighborhoods is compact.

spaces/S000174/properties/P000051.md

@@ -0,0 +1,7 @@
+---
+space: S000174
+property: P000051
+value: false
+---
+
+The subspace $2 ^ \omega \setminus \{(0,0,\dots)\} \subseteq X$ is homeomorphic to {S26} without a point, and so has no isolated points.

spaces/S000174/properties/P000062.md

@@ -0,0 +1,10 @@
+---
+space: S000174
+property: P000062
+value: false
+---
+
+The collection $\{\xi^\omega : \xi < \omega_1\}$ is an open cover of $X$.
+Every countable subcollection has union contained in $\xi^\omega$ for some
+$\xi < \omega_1$, and $\xi^\omega$ is a proper closed subset of $X$. Thus,
+no countable subcollection has dense union.

spaces/S000174/properties/P000065.md

@@ -0,0 +1,9 @@
+---
+space: S000174
+property: P000065
+value: true
+---
+
+
+\[\mathfrak c = |2^\omega| \leq |\omega_1^\omega|
+\leq |(2^\omega)^\omega| = \mathfrak c\]

spaces/S000174/properties/P000083.md

@@ -0,0 +1,16 @@
+---
+space: S000174
+property: P000083
+value: false
+refs:
+  - doi: 10.4064/FM-97-2-53-55
+    name: "A perfectly normal locally metrizable non-paracompact space"
+  - zb: "0593.28016"
+    name: Borel measures (Gardner and Pfeffer)
+---
+
+Proposition 2 of {{doi:10.4064/FM-97-2-53-55}} constructs a locally countable
+subspace of $X$ which is not $\sigma$-discrete. By Theorem 13.3 of
+{{zb:0593.28016}}, every locally countable subspace of
+a weakly $\delta\theta$-refinable space is $\sigma$-discrete. Hence $X$ is not
+weakly $\delta\theta$-refinable and, in particular, is not meta-Lindelöf.

spaces/S000174/properties/P000093.md

@@ -0,0 +1,7 @@
+---
+space: S000174
+property: P000093
+value: false
+---
+
+The point $(\omega, \omega,\dots) \in X$ has no countable neighborhood.

spaces/S000174/properties/P000139.md

@@ -0,0 +1,7 @@
+---
+space: S000174
+property: P000139
+value: true
+---
+
+The point $p := (0, 0, \dots) \in X$ is isolated since $\{p\} = 1^\omega$ where $1 = \{0\}$.

spaces/S000174/properties/P000194.md

@@ -0,0 +1,16 @@
+---
+space: S000174
+property: P000194
+value: false
+refs:
+  - doi: 10.4064/FM-97-2-53-55
+    name: "A perfectly normal locally metrizable non-paracompact space"
+  - zb: "0593.28016"
+    name: Borel measures (Gardner and Pfeffer)
+---
+
+Proposition 2 of {{doi:10.4064/FM-97-2-53-55}} constructs a locally countable
+subspace of $X$ which is not $\sigma$-discrete. By Theorem 13.3 of
+{{zb:0593.28016}}, every locally countable subspace of
+a weakly $\delta\theta$-refinable space is $\sigma$-discrete. Hence $X$ is not
+weakly $\delta\theta$-refinable and, in particular, is not submetacompact.